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[Katsunori Wakabayashi](https://orcid.org/0000-0002-9147-9939)

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[Hierarchical disorder in moiré exciton photoluminescence probed by spectral-descriptor correlations](https://mdr.nims.go.jp/datasets/067d9fb0-4670-4f3d-9399-01675112b8d6)

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Hierarchical Disorder in Moiré Exciton Photoluminescence Probed bySpectral-Descriptor CorrelationsKatsunori Wakabayashi∗Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba 305-0044, Japan(Dated: July 18, 2026)Hyperspectral photoluminescence (PL) maps of moiré transition-metal dichalcogenide heterobi-layers encode rich information about the underlying disorder, but extracting it is hampered by theambiguity of assigning individual spectral peaks. We show that the spatial correlations amonga set of simple, peak-decomposition-free spectral descriptors—such as the centroid energy, thedominant-peak energy, and a sharp-line fraction—provide a peak-decomposition-free route to in-fer features of that disorder. Different descriptors act as filters that select different componentsof a multi-scale disorder landscape—a smooth, micron-correlated background and a dense set oflocalized traps—and therefore acquire different spatial correlation lengths. The central prediction isa correlation-length hierarchy, ξ(Ecent) ≥ ξ(Edom), which we derive by splitting the dominant-peakenergy into a smooth background part and a short-range trap-switching fluctuation. The same pic-ture explains the measured inter-descriptor correlations, including the near-perfect anticorrelationρS(∆Ecd, RHL) ≈ −0.978, which we show to be a robust geometric trend for spectra dominatedby a common emission-envelope asymmetry. Benchmarked against phenomenological simulations,Hamiltonian diagonalization, and the measured MoSe2/WSe2 descriptor correlations, the frameworkturns descriptor maps into a quantitative, peak-decomposition-free probe of slow disorder and localtraps in moiré excitons and, more broadly, in disordered semiconductor emitters.I. INTRODUCTIONMoiré heterostructures formed by vertically stackingtwo-dimensional materials with a small twist angle or lat-tice mismatch have become a central platform for engi-neering quantum states of electrons and excitons [1–8]. Intransition metal dichalcogenide (TMD) heterobilayers—building on the strong photoluminescence of monolayerTMDs [9–11] and their rich optical and valley physics[12–14]— the long-wavelength moiré potential reorga-nizes interlayer excitons into localized minibands, en-abling moiré-trapped exciton physics including moiréBose-Hubbard models, exciton crystals, and quantumemitter arrays [15–25].Among these systems, MoSe2/WSe2 heterobilayers areparticularly rich platforms for studying moiré-modulatedoptical responses. At low temperatures, the interlayer ex-citon photoluminescence (PL) is characteristically com-plex: a broad emission envelope near 1.25–1.40 eV coex-ists with dense, narrow spectral peaks whose microscopicassignment remains debated, with proposed contribu-tions from moiré-trapped interlayer excitons, phonon-assisted momentum-indirect emission, donor-acceptorpair excitons, and disorder-localized states [17, 26–34].A key observation motivating the present work is thatthe spectral complexity is spatially organized. Figure 1sketches the system: a twisted MoSe2/WSe2 moiré heter-obilayer [Fig. 1(a)], its hyperspectral PL map [Fig. 1(b)],and the multi-scale disorder landscape underlying ourmodel [Fig. 1(c)].∗ WAKABAYASHI.Katsunori@nims.go.jpRecent hyperspectral PL mapping experiments by Ah-mad et al. [35] on a MoSe2/WSe2 heterobilayer re-veal that nine peak-decomposition-free spectral descrip-tors (centroid energy Ecent, dominant-peak energy Edom,centroid-dominant offset ∆Ecd, quantile width W80,high/low ratio RHL, sharp fraction FS , roughness R1,spectral entropy Sspec, and integrated intensity Itot) showcharacteristic micron-scale spatial correlations. Feature-wise spatial correlation analysis yields 1/e decay lengthsof 1.27–2.05 µm for all descriptors, exceeding the 0.85 µmoptical spot size. Gaussian mixture clustering on theprincipal component analysis (PCA)-projected descrip-tor space identifies three dominant spectral families thatform contiguous real-space domains. At the same time,individual pixels retain a dense multi-peak structure thatvaries across the map, implying an unresolved local spec-tral manifold below optical resolution.Here, the local spectral manifold denotes the opticallyunresolved ensemble of local emitting states—includingmoiré-localized and trap-related exciton states—that col-lectively form the PL spectrum within a single opticalspot; we use the term in this physical sense, not in thedifferential-geometric one. More broadly, hyperspectralPL imaging is emerging as a spatially resolved probe ofstrain and disorder in TMD systems [36].This combination of phenomena—resolved micron-scale spectral landscape coexisting with unresolved local-spectral-manifold complexity—suggests that a single dis-order scale is insufficient. The central experimental con-clusion is that the emission is organized hierarchically :a slowly varying micron-scale background shapes the en-velope, while local spectral manifold degrees of freedomproduce dense fine structure within that background.Despite extensive experimental and numerical studiesmailto:WAKABAYASHI.Katsunori@nims.go.jp2θMoSe2WSe2twistangle θ            (a) MoSe2/WSe2 moiré heterobilayer (b) Hyperspectral PL mapping (c) Multi-scale disorder landscapeLowHighDescriptor extractionSlow-varyingpotentialLocal traps +Total potential=1.2 1.41.30-10+10meVFIG. 1. Overview of the system, measurement, and theoretical framework. (a) MoSe2/WSe2 heterobilayer withinterlayer distance ∼ 0.7 nm and twist angle θ, producing a long-wavelength moiré superlattice of period aM . (b) Hy-perspectral photoluminescence mapping yields a spatially resolved PL spectrum at each map position (x0, y0), from whichpeak-decomposition-free spectral descriptors (Ecent, Edom, ∆Ecd, W80, RHL, FS , R1, Sspec, Itot) are extracted without mi-croscopic line assignment. (c) Theoretical multi-scale disorder landscape: a slowly varying background potential Vslow(r) atthe micron scale combined with localized traps Vtrap(r) at sub-optical length scales yields the micron-scale effective potentialVeff(r) ≃ Vslow(r) + Vtrap(r) that organizes the spectral descriptors hierarchically; the moiré term in the full potential [Eq. (2)]is coarse-grained over the optical spot and enters only through the renormalized mass and local density of states.of moiré exciton physics, a minimal theoretical frame-work for hierarchical spectral inhomogeneity as an orga-nizing principle has not been established. Previous theo-retical work has focused primarily on: (i) single-particlemoiré miniband structures [37–41], (ii) correlated excitonand electron phases [19, 20, 42–44], (iii) isolated defect-trapped excitons as quantum emitters [45–47], or (iv)Gaussian disorder models for energy broadening withoutspatial structure [28]. These approaches do not directlyaddress the separation of scales observed in hyperspec-tral mapping.In this paper, we develop a minimal theory for the spa-tial organization of spectral descriptors in moiré excitonPL—minimal in the sense of containing the fewest ingre-dients needed to reproduce the observed descriptor hier-archy, not minimal in a microscopic sense—built arounda single organizing principle: different descriptors actas filters for different components of the disorder land-scape. A descriptor that integrates spectral weight (cen-troid energy) averages over local trap fluctuations andtracks the smooth, micron-scale potential Vslow, whilea descriptor that locates spectral peaks (dominant-peakenergy) is additionally randomized by trap-level switch-ing within the optical spot. This differential filtering iswhy the descriptor-specific correlation lengths differ, whythe inter-descriptor Spearman matrix has the structureit does, and why three spectral families emerge from acontinuous disorder landscape. It thereby connects theleading effective disorder parameters (ξs,Ws,Wt) to theobservable descriptor covariance structure, without anymicroscopic peak assignment.The principal contributions of this work are the fol-lowing. (i) We formalize the descriptor-based disorder-filter principle introduced above into a quantitative map-ping from the descriptor covariance matrix to the disor-der parameters (ξs,Ws,Wt) [48]. (ii) We derive a cor-relation length hierarchy ξ(Ecent) ≥ ξ(Edom) fromthe decomposition Edom = V̄s + δEdom. The trap-switching fluctuation δEdom adds short-range variance,which systematically shortens the correlation length ofthe dominant-peak energy relative to that of the cen-troid. (iii) We show that the nontrivial content of thishierarchy lies not in the mere fact that the centroid issmoother than the peak position (expected of any aver-age), but in its magnitude: the ratio ξ(Ecent)/ξ(Edom)quantitatively encodes the relative strength of the trapand slow disorder and the trap density. We turn thisinto an explicit, peak-decomposition-free protocol (Ta-ble V) that constrains the leading effective disorder pa-rameters from four measured descriptor observables—the centroid correlation length, the centroid variance,the Ecent–Edom correlation, and the mean sharp frac-tion (the last fixing the trap density only after calibra-tion of the sharp-fraction scale)—applicable, in princi-ple, to hyperspectral PL maps containing both smoothenvelopes and unresolved local fine structure. (iv) We3derive analytically the sign and approximate magnitudeof the principal inter-descriptor Spearman correlations.This establishes the ∆Ecd–RHL anti-correlation as a ro-bust spectral-shape relation, valid for a broad class ofspectra with a dominant unimodal envelope, and showsthe FS–R1 co-variation to follow from both descriptorsmeasuring the same trap-peak density. (v) We pro-vide an independent model-level check via Hamiltoniandiagonalization—without relying on the synthetic two-fluid spectrum construction—recovering the correlation-length hierarchy from the eigenvalue spectrum alone. (vi)We construct a disorder parameter map that identifiesfour qualitatively distinct regimes and places the exper-imental system in the hierarchically disordered regimewhere both slow and trap disorder exceed the homoge-neous linewidth.Throughout, the theoretical framework is a spectral-statistical organization theory : the Hamiltonian defines acontinuum spectral manifold whose coarse-grained statis-tics are resolved by the optical spot, and the descriptorsare the observables through which the disorder hierar-chy is encoded and constrained. This is distinct from aquantum-transport or localization theory—the relevantphysics operates at the ∼ µm scale of the optical enve-lope, not at the nanometer scale of individual eigenstates.The remainder of this paper is organized as follows.Section II introduces the effective Hamiltonian and dis-order model. Section III develops the Green’s functionformalism and local PL spectrum. Section IV derives thestatistical properties of spectral descriptors as disorderfilters. Section V derives the correlation length hierarchyrelation. Section VI derives inter-descriptor Spearmancorrelations. Section VII maps the four disorder regimes.Section VIII presents the local-density-of-states (LDOS)two-fluid structure. Section IX interprets spectral fami-lies as correlated disorder domains, and Section X charac-terizes the multi-scale spectral organization that emergesin the hierarchical regime. Section XI describes Hamilto-nian diagonalization and phenomenological simulations.Section XII discusses limitations and parameter extrac-tion. Section XIII concludes.II. MODEL: EFFECTIVE EXCITONHAMILTONIANThis section constructs the effective Hamiltonian onwhich the rest of the paper rests. We model the in-terlayer exciton as a single center-of-mass particle mov-ing in an effective potential, and assemble that poten-tial from three physically distinct ingredients—the moirésuperlattice, a slowly varying long-wavelength disorder,and sparse local traps—each introduced in turn below.We then identify the separation of length scales amongthese ingredients, which is what makes the descriptor-filter analysis of the following sections tractable.A. Center-of-Mass HamiltonianWe describe the interlayer exciton by its center-of-mass(COM) coordinate r in the moiré plane. The internalexciton structure is integrated out, yielding an effectivesingle-particle problem for the COM motion:H = − ℏ22M∇2 + Veff(r), (1)where M is the interlayer exciton effective mass. ForMoSe2/WSe2, the electron and hole are spatially sepa-rated into different layers [21, 23], givingM ≈ me+mh ≈1.0–1.5m0 [12, 49, 50] wherem0 is the free electron mass.The effective potential is decomposed into three com-ponents acting on different length scales:Veff(r) = Vmoiré(r) + Vslow(r) + Vtrap(r). (2)These three components describe physics at three distinctlength scales, as summarized in Table I.B. Moiré PotentialThe moiré superlattice potential is modeled by thelowest-order hexagonally symmetric Fourier components:Vmoiré(r) = 2Vm3∑j=1cos(Gj · r+ ϕj), (3)where Gj are the three inequivalent moiré reciprocallattice vectors satisfying |Gj | = 4π/(aM√3), Vm con-trols the modulation amplitude, and ϕj encodes the lo-cal stacking registry. For small twist angle θ and latticeconstant a0, the moiré period is aM ≈ a0/θ.For a twist angle θ ∼ 2–4◦, aM ∼ 10–20 nm, and themoiré potential depth is estimated at Vm ∼ 10–50 meVfrom first-principles calculations [37–40, 51].This term confines excitons to moiré unit cells, creatinglocalized miniband-like states on the nanometer scale.For the purposes of spatial PL statistics at the micronscale, the moiré potential can be treated as a fast-varyingbackground that renormalizes the exciton effective massand density of states (DOS), but does not directly set themicron-scale spatial correlations.C. Long-Wavelength Correlated DisorderThe dominant component governing micron-scale spec-tral organization is a slowly varying, spatially correlatedrandom potential:⟨Vslow(r)⟩ = 0, ⟨Vslow(r)Vslow(r′)⟩ =W 2s Cs(|r− r′|),(4)4Component Length Scale Physical Origin RoleVmoiré aM ∼ 10–20 nm Moiré superlattice Miniband formationVslow ξs ∼ 1–2 µm Strain, twist-angle gradient Spectral domain formationVtrap σt ≲ σopt (few–100 nm effective) Localized traps (defects/strain/charge) Sharp line generationTABLE I. Three disorder components and their characteristic length scales. Here σt is an effective trap width spanning theexperimentally debated sharp-emitter origins (atomic defects, strain, or electrostatic disorder), not a microscopic defect size;the theory requires only σt ≲ σopt ≪ ξs.with Gaussian correlation kernel:Cs(r) = exp(− r22ξ2s). (5)The characteristic parameters are: Ws ∼ 5–20 meV (dis-order amplitude) and ξs ∼ 1–2 µm (correlation length),much larger than the moiré period aM .The physical origins of Vslow are diverse. Spatial vari-ations in the local twist angle δθ(r), directly imaged intwisted TMD bilayers [52, 53], shift the interlayer ex-citon energy as ∂E/∂θ · δθ (the twist-angle dependenceof interlayer-exciton properties in TMD heterobilayers iswell established [29]). Inhomogeneous strain fields εij(r)couple through the deformation potential ∼ Dijεij [54].Electrostatic potential fluctuations arise from trappedcharges or dielectric inhomogeneity [55]. Finally, moiréreconstruction domain walls introduce step-like modula-tions [52, 53, 56]. All of these generically produce smoothfields with correlation lengths set by the macroscopicsample quality, typically 1–5 µm in high-quality van derWaals assemblies.We work in the Gaussian approximation (4) as the min-imal model, realized numerically by the spectral (FFT)method of Appendix A; extensions to non-Gaussian slowdisorder are discussed in Section XII.D. Local Trap DisorderLocal trap potentials are introduced as a sum of at-tractive Gaussians centered at random positions {Ri}:Vtrap(r) =∑iui exp(−|r−Ri|22σ2t), (6)where Ri are drawn from a Poisson point process withdensity nt; ui ≤ 0 are random trap depths drawn froma distribution P (u) with mean ū < 0 and characteristicdepth scale Wt (the simulations realize P (u) as a half-normal, ui = −|N (0,Wt)|, so that Wt is the scale of theunderlying normal, with second moment ⟨u2⟩ =W 2t andvariance W 2t (1− 2/π)); and σt ≪ ξs is the trap radius.Each trap acts as a local potential minimum that canbind an exciton and produce a sharp spectral line. Thetrap correlation function is:⟨Vtrap(r)Vtrap(r′)⟩ = nt⟨u2⟩·πσ2t exp(−|r− r′|24σ2t), (7)with correlation length ∼√2σt ≪ ξs. This spa-tial white-noise-like character of the trap disorder iswhat distinguishes it from Vslow. Microscopic sourcesof such localized potentials include native point defects[45, 57], strained nanobubbles and strain-localized emit-ters [31, 47], and adsorbate-induced potential wells.E. Parameter Hierarchy and Scale SeparationThe effective model assumes a clear separation oflength scales,σt, aM ≪ ξs ≪ L, (8)where L is the sample (or map) size. In the experi-ment [35], L ∼ 8 µm and ξs ∼ 2 µm. The two mi-croscopic scales of the model lie far below ξs: the moiréperiod aM ∼ 10–20 nm, and the trap radius σt, whichranges from the atomic/defect scale (a few nm) up to∼100 nm in our robustness tests.We deliberately leave the relative ordering of σt andaM unspecified. Instead, σt is treated as an effectivepotential width rather than a microscopic trap dimen-sion, because the origin of the sharp emitters—moiré-trapped excitons, atomic defects, or strain and electro-static disorder—remains experimentally debated [26, 28].These possible origins have characteristic sizes rangingfrom a few nm up to ∼100 nm, so no single microscopicvalue of σt is preferred.The theory requires only σt ≲ σopt ≪ ξs (with σopt theoptical-spot radius), so that each trap acts as a sub-spotlocalized emitter, with many such emitters contributingto a single measured spectrum. With this separation, σtenters the descriptor statistics only through the productntW2t σ2t . This combination is not arbitrary: it is the vari-ance of the trap potential Vtrap. Summing the squaredcontribution of each well over the Poisson-distributedtrap positions gives the shot-noise variance (Campbell’stheorem [58, 59]) VarVtrap ∝ ntW2t σ2t , with the three fac-tors being the areal density nt, the depth variance W 2t ,and the well area ∼ σ2t . We identify it as the effective5trap disorder Γt in Sec. VII, and the descriptor variancesthat depend on it are derived in Appendix C.Because σt appears only inside this product, its valueis not pinned down on its own: any change in σt canbe offset by a compensating change in the trap density(nt ∝ σ−2t ), or in the depth variance W 2t , that keepsthe product—and hence the leading trap-loading contri-bution to the descriptor variances—approximately un-changed. In other words, σt, nt, and W2t are degenerate:they enter only in this one combination, so the predic-tions are insensitive to the precise value of σt within theallowed window. The only place σt sets an independentlength scale is the hierarchy floor ξt ≡√2σt of Eq. (38);even for σt ∼ 100 nm this remains well below ξs (ξt/ξs ∼0.07) and therefore does not bind the hierarchy. Thecorrelation-length hierarchy and the inter-descriptor cor-relations are thus controlled primarily by (ξs,Ws,Wt, nt),with only weak dependence on σt within the sub-opticalrange; a diagonalization sweep over σt = 50–125 nm con-firms this explicitly (Appendix D).This separation (8) justifies treating the disorder com-ponents as statistically independent and analyticallytractable at each scale. Figure 2 shows a representa-tive realization of the resulting hierarchical disorder land-scape: the slow potential Vslow [Fig. 2(a)], the local trappotential Vtrap [Fig. 2(b)], and the effective potentialVeff = Vslow + Vtrap [Fig. 2(c)].III. GREEN’S FUNCTION FORMALISM ANDLOCAL PL SPECTRUMThe spectral descriptors introduced above are all com-puted from the local PL spectrum I(E,R) — the emis-sion collected from an optical spot centered at positionR. The goal of this section is to express that observ-able directly in terms of the disorder Hamiltonian H ofSec. II, so that the statistics of the descriptors can laterbe traced back to the underlying disorder parameters.The retarded Green’s function is the natural tool forexpressing I(E,R) in terms of H. It resolves H into itsspectral content, and its imaginary part yields the localdensity of states (LDOS): the spectral weight that theeigenstates of H place at energy E and position r. Themeasured local spectrum is then this LDOS filtered bythe finite optical spot and weighted by the radiative cou-pling of each eigenstate to light. In this way the formal-ism converts the eigenstates {ψn, En}, which encode themulti-scale disorder, into the observable I(E,R) with-out fitting or identifying individual spectral peaks — thepeak-decomposition-free approach on which the descrip-tor analysis is based. We build this chain in three steps:the Green’s function, the LDOS it generates, and theradiatively weighted, optically filtered local spectrum.A. Retarded Green’s FunctionThe retarded Green’s function is the resolvent of theexciton Hamiltonian: its poles sit at the eigenenergies Enand its residues carry the eigenfunctions ψn, so it pack-ages the entire spectral content of H in a single object[60],G(r, r′;E) = ⟨r|(E + iη −H)−1|r′⟩ =∑nψn(r)ψ∗n(r′)E − En + iη,(9)where ψn(r) and En are the eigenstates and eigenenergiesof H, and η → 0+ is an infinitesimal broadening.B. Local Density of StatesThe local density of states (LDOS) is:ρ(E, r) = − 1πImG(r, r;E) =∑n|ψn(r)|2 δ(E − En).(10)In practice, the delta function is replaced by a Lorentzianor Gaussian with linewidth η ∼ 1–3 meV (phenomeno-logical exciton lifetime broadening):ρη(E, r) =∑n|ψn(r)|2 gη(E − En), gη(E) =η/πE2 + η2.(11)C. Local PL SpectrumThe connection to experiment is set by the radiativecoupling of the center-of-mass (COM) exciton eigenstatesto light. Because the internal 1s relative wave func-tion is essentially rigid, the light couples to each COMeigenstate ψn(r) through its coherent dipole (oscillator-strength) matrix element, the amplitude integral of theCOM wave function over the emission region, so that thefar-field radiative weight of eigenstate n is ∝ |∫ψn d2r|2rather than the incoherent probability density |ψn|2; thisis the standard result for disorder-localized exciton emis-sion in quantum wells and related nanostructures [61–63]. For a local measurement that collects from an op-tical spot centered at R with amplitude collection modeWamp, the locally detected PL spectrum isI(E,R) =∑nfocc(En)An(R) gη(E − En), (12)with the coherent radiative weightAn(R) = |Mn(R)|2, Mn(R) =∫Wamp(r−R)ψn(r) d2r,(13)60 2 4 6 8x ( m)02468y (m)(a) Vslow(r)Ws = 6 meVs = 2 m0 2 4 6 8x ( m)02468(b) Vtrap(r)Wt = 12 meV, nt = 2 m 2Nt = 133 traps (red ×)0 2 4 6 8x ( m)02468(c) Veff = Vslow + Vtrapcombined landscape1050510meV43210meV1050510meVFIG. 2. Hierarchical disorder landscape. (a) Slow correlated disorder Vslow(r) with amplitude Ws = 6 meV andcorrelation length ξs = 2 µm, generated by FFT-based Gaussian filtering. The smooth red/blue domains of size ∼ ξs representthe slow-disorder component that gives rise to micron-scale spectral families in the model. (b) Trap disorder Vtrap(r) consistingof Nt = 133 Gaussian potential wells (Wt = 12 meV, σt = 50 nm, nt = 2 µm−2, consistent with ntL2 ≈ 128 over the 8× 8 µm2domain). The color scale is saturated at the 2nd percentile of Vtrap for visibility; because the field is near zero between the sparsewells, this displayed lower bound is much shallower than the deepest individual well centers, whose bare trap-depth distributionis parameterized by Wt = 12 meV. Red crosses mark the trap centers, which are drawn from P (r) ∝ exp(−Vslow(r)/Ebias) withEbias = 7 meV, biasing the trap positions, or optically active trap sites, toward slow-potential minima. (c) Effective potentialVeff = Vslow + Vtrap, showing the combined landscape experienced by excitons. The gold dashed rectangle indicates the 20× 20measurement grid area (spanning 7.6 × 7.6 µm2, i.e. a 0.4 µm pitch matching the experimental 400 nm step, with a 0.2 µmmargin from each edge).where Wamp is the (real, positive) collection-mode am-plitude andWopt = |Wamp|2 is the intensity point-spreadfunction. For a diffraction-limited Gaussian spot bothare Gaussian,Wamp(r) =1√2πσ2optexp(− |r|24σ2opt), (14)Wopt(r) = |Wamp(r)|2 =12πσ2optexp(− |r|22σ2opt), (15)so that Wamp has width√2σopt and the normalized in-tensity PSFWopt has width σopt = FWHM/(2√2 ln 2) ≃0.36 µm (experimental FWHM = 0.85 µm). The occu-pation factor is focc(E). Only the coarse-graining scaleσopt enters the descriptor correlations; the overall nor-malization sets only the absolute emission intensity. Thecoherent weight (13) preferentially enhances nodeless lo-calized states and suppresses spatially extended stateswith sign-changing wave functions. The mechanism is thecancellation in the amplitude integral Mn =∫Wamp ψn:a nodeless, well-localized trap state keeps a single signacross its small extent, so its contributions add up co-herently and An is large; an extended state whose wavefunction oscillates in sign has its positive and negativecontributions largely cancel, so An is small. Thus, inthis approximation, localized trap-like states carry largerradiative weights, whereas extended background statestend to be suppressed by phase cancellation—a PL-levelanalogue of the mobility-edge selectivity known for dis-ordered quantum-well excitons [61, 62].For the analytic descriptor theory it is convenient touse the incoherent local-density approximation, in which|Mn(R)|2 is replaced by∫d2rWopt(r − R) |ψn(r)|2, sothat Eq. (12) becomes the optically filtered LDOSI(E,R) =∫d2rWopt(r−R) focc(E) ρη(E, r). (16)Because both weights are localized within the opticalspot, the descriptor covariance and the correlation-lengthhierarchy derived below are essentially the same for thecoherent and incoherent forms. We therefore use thetransparent LDOS form (16) both in the analytic treat-ment and in the microscopic diagonalization of Sec. XID,and verify in Appendix E that replacing it by the micro-scopically motivated coherent radiative form (13) leavesthe descriptor statistics unchanged within the numeri-cal seed-to-seed uncertainty (the two weightings differ by≤ 0.01 for every descriptor).Approximating focc ≈ 1 for all states in the analyzedPL window yields the working formI(E,R) =∑nAn(R) gη(E − En). (17)Here An(R) denotes the local optical weight in eitherrepresentation: the coherent radiative form of Eq. (13),or the incoherent local-density approximation An(R) =7∫d2rWopt(r − R) |ψn(r)|2 [Eq. (16)] used in the ana-lytic treatment below. A full treatment would fix foccfrom the relaxation kinetics between disorder eigenstates[62], which at low temperature can be strongly energyselective—for example through a mobility edge—andwhich we do not attempt to model here. Instead, weadopt the constant-occupation form focc ≃ 1 as a delib-erate minimal (null) model, not as a thermal-equilibriumapproximation, so that the descriptor-correlation hier-archy is attributed to the disorder-filtering mechanismrather than to occupation-induced weighting. Withinthis minimal model, a spatially uniform, smooth energy-dependent occupation would mainly reshape the com-mon spectral envelope over the analyzed window (∼150 meV, i.e. 1.25–1.40 eV) and is not expected tochange the spatial correlation hierarchy, since the lat-ter is set by the spatial statistics of the eigenenergies Enand weights An(R). A strongly energy-selective occu-pation, or mobility-edge filtering of the kind emphasizedin Refs. [61, 62], would constitute an additional physi-cal weighting beyond the present minimal model, whichwe note here as a limitation. The optical spot intensityfunction is modeled as:Wopt(r) =12πσ2optexp(− |r|22σ2opt), (18)with σopt = FWHM/(2√2 ln 2) ≈ 0.36 µm for a fullwidth at half maximum (FWHM) of the optical spot of0.85 µm.D. Spectral Decomposition: Background and TrapContributionsA key analytical insight is to decompose I(E,R) intocontributions from states with different spatial extents.We classify eigenstates using the inverse participation ra-tio (IPR), which provides a standard measure of real-space localization [61] and thus naturally separates spa-tially extended background states (small IPR) from trap-localized states (large IPR). We define the localizationlength ℓn = IPR−1/2n , where IPRn =∫d2r |ψn(r)|4 (withψn normalized). The IPR works as an inverse effectivearea: for a state spread uniformly over an area A one has|ψn|2 ∼ 1/A and IPRn ∼ 1/A, so ℓn = IPR−1/2n mea-sures the linear extent over which the state has appre-ciable weight—small ℓn for tightly localized trap states,large ℓn for extended background states [61]. A cutoff ℓ∗(of order the trap size) then separates the two families:I(E,R) = Ibg(E,R) + Isharp(E,R), (19)Ibg(E,R) =∑n: ℓn>ℓ∗An(R) gη(E − En), (20)Isharp(E,R) =∑n: ℓn≤ℓ∗An(R) gη(E − En), (21)where ℓ∗ is a crossover localization length (see Sec-tion VIII). States with ℓn > ℓ∗ are spatially extendedon the scale of the optical spot and contribute to thesmooth background envelope Ibg. Trap-concentratedstates (ℓn ≤ ℓ∗) contribute narrow lines to Isharp. TheIPR here serves to define this conceptual two-componentdecomposition, with ℓ∗ acting as a practical crossoverrather than a sharp microscopic phase boundary. In thestrongly disordered regime of the numerical simulations,where the IPR is set by the disorder landscape ratherthan by quantum interference, we apply an equivalentenergy-based classification of the same two families (Sec-tion VIII).E. Optical Coarse-Graining of Slow DisorderThe slow disorder component enters the measured PLspectrum primarily through optical coarse-graining. Theexperimentally measured spectrum at position R is notthe LDOS at a single microscopic point, but the LDOSaveraged over the optical point-spread function Wopt(r−R) [Eq. (17)]. Since Vslow(r) varies on the length scale ξs,much larger than the optical spot size σopt (σopt/ξs ≪ 1),the smooth part of the local spectral manifold withinone optical spot is shifted almost uniformly by the localaverage of Vslow.We therefore define the optically averaged slow poten-tial asV̄s(R) =∫d2rWopt(r−R)Vslow(r). (22)This quantity is the energy shift of the background PLenvelope observed at the optical position R, rather thanthe first-order energy correction of a single microscopiceigenstate; it requires no assumption about the detailedprofile of individual wavefunctions.Equivalently, for the background contribution to theLDOS one may writeρbg(E,R) ≃ ρ(0)bg(E − V̄s(R)), (23)up to corrections controlled by the small ratio σopt/ξs.This relation expresses the central coarse-graining as-sumption used below: the smooth background within oneoptical spot is rigidly displaced by V̄s(R), so that centroid-like descriptors, which integrate spectral weight over thesmooth envelope, primarily track V̄s(R), whereas peak-position descriptors remain sensitive to additional localtrap-level fluctuations within the same optical spot.IV. STATISTICAL THEORY OF SPECTRALDESCRIPTORSThis section makes the disorder-filter principle quan-titative. We first define the spectral descriptors—simple, peak-decomposition-free functionals of the local8spectrum—and then derive the statistics of the most im-portant ones, showing explicitly that the centroid energytracks the slow disorder while the spectral roughness andentropy track the traps. These results are the analyticalbasis for the correlation-length hierarchy and the inter-descriptor correlations derived in the following sections.A. Definitions of Spectral DescriptorsWe define the full set of spectral descriptors follow-ing the operational conventions of the experiment [35],so that simulated and measured descriptors are directlycomparable. Each descriptor is a simple functional ofthe raw local spectrum I(E,R) that requires no peakfitting or microscopic line assignment. The set is de-liberately chosen so that different descriptors are sensi-tive to different aspects of the spectral shape—its overallposition, its width, its asymmetry about the dominantpeak, and its fine structure—and therefore, through thedisorder-filter principle, to different components of theunderlying disorder: descriptors that integrate the wholespectrum (such as the centroid) track the smooth back-ground, whereas peak- and roughness-based descriptorsrespond to the sharp, trap-generated fine structure. Weintroduce them in turn, in each case stating what thedescriptor measures before giving its definition.1. Centroid EnergyThe centroid is the intensity-weighted mean emissionenergy—the first moment of the local spectrum. Becauseit integrates over the entire spectrum, it averages outthe sub-spot trap fluctuations and tracks the smooth,slowly varying background set by Vslow; it is therefore thedescriptor most directly tied to the micron-scale disorder(Sec. V). It is defined byEcent(R) =∫E I(E,R) dE∫I(E,R) dE. (24)2. Dominant-Peak EnergyThe dominant-peak energy is the position of the tallestspectral feature—the mode of the local spectrum. In con-trast to the centroid, it is set by whichever single statecarries the most optical weight within the optical spot,so it responds to discrete trap-level switching as the spotis moved and, as shown below, acquires a shorter corre-lation length than Ecent. It is defined byEdom(R) = argmaxEI(E,R). (25)Here, argmaxE denotes the energy at which the local PLspectrum reaches its maximum; for discretized spectra,it is taken as the energy bin with the largest intensity.3. Centroid-Dominant OffsetThe offset between the centroid and the dominant peakmeasures how far the intensity-weighted mean lies fromthe mode, i.e. the skewness of the emission envelopeabout its own maximum. It vanishes for a symmetricline and takes a definite sign as spectral weight is re-distributed to the low- or high-energy side of the domi-nant peak, making it a sensitive, peak-decomposition-freeprobe of envelope asymmetry. It is defined by∆Ecd(R) = Ecent(R)− Edom(R). (26)4. Quantile WidthThe quantile width characterizes the overall energyspread of the emission without reference to any individ-ual peak. Using the central-80% band rather than a vari-ance or a fitted linewidth makes it robust to weak far tailsand to the multi-peak structure of the local spectrum, sothat it reflects the width of the dominant emission band.Specifically, W80(R) is the energy width of the band con-taining the central 80% of the integrated PL intensity,i.e. the interval between the 10th and 90th cumulative-intensity percentiles.5. High/Low RatioThe high/low ratio quantifies the asymmetry of theemitted intensity about the dominant peak—how muchspectral weight lies below the main line relative to aboveit. It captures the same envelope skew as the offset∆Ecd, but through the distribution of intensity ratherthan of energy, which is why the two turn out to be sotightly linked (Sec. VI). Following the experimental con-vention [35], the spectrum is split at the dominant peakEdom (not at a fixed global energy):RHL(R) =∫ Edom−∞ I(E,R) dE∫∞EdomI(E,R) dE, (27)so that RHL > 1 when spectral weight accumulates belowthe dominant emission peak. This shared Edom referenceis essential: it is what makes RHL and ∆Ecd two mea-sures of the same envelope asymmetry about the domi-nant peak, and hence tightly anticorrelated. Referencingthe split to Ecent or to a fixed spectral quantile insteadmeasures a different quantity and weakens or reverses therelation, as expected.6. Sharp FractionThe sharp fraction is the fraction of the emitted in-tensity carried by narrow, trap-related lines rather than9by the broad background envelope. It is a direct, peak-decomposition-free proxy for the local trap loading—thenumber and depth of sharp emitters within the opticalspot—and is therefore the descriptor most sensitive tothe trap component of the disorder. It is defined byFS(R) =Isharp,tot(R)Itot(R), (28)where Isharp,tot =∫Isharp(E,R) dE is the intensity car-ried by sharp spectral features. Operationally, sharppeaks are identified by local-maximum detection: a localmaximum qualifies as sharp if its prominence exceeds afraction p of the pixel maximum (p = 0.02) and its widthat half-prominence is below a threshold w (w = 10 meV),which selects narrow trap lines while excluding the broadbackground envelope (FWHM ∼ 35 meV); Isharp is theintensity within±w/2 of the qualifying peaks. This peak-detection criterion—rather than a fixed linewidth thresh-old or a parametric background subtraction—is robust toits parameters: varying p over 0.01–0.05 and w over 8–15 meV changes ρS(FS , R1) by less than 0.03 (from +0.90to +0.92), while the mean ⟨FS⟩ shifts only within 0.25–0.39.7. Spectral RoughnessThe spectral roughness is the total variation of the nor-malized spectrum—its integrated absolute slope—and soquantifies the amount of sharp structure it contains:a smooth envelope gives a small R1, while each addi-tional narrow trap line adds to the total variation. It isthus a peak-detection-free proxy for the density of finestructure, complementary to the sharp fraction FS (withwhich it is strongly correlated, Sec. VI). Normalizing bythe integrated intensity makes it independent of the ab-solute emission strength:R1(R) =∫|dI(E,R)/dE| dE∫I(E,R) dE. (29)8. Spectral EntropyThe spectral entropy measures how uniformly theemitted intensity is distributed across energy, treatingthe normalized spectrum as a probability distribution. Itis small when the emission is concentrated in a few fea-tures and approaches unity for a broad, feature-rich spec-trum, and it therefore provides a peak-decomposition-free measure of local spectral complexity that comple-ments the quantile width. Normalized by lnN to lie in[0, 1], it readsSspec(R) = − 1lnN∑kpk(R) ln pk(R), (30)where pk = I(Ek,R)/∑j I(Ej ,R) is the normalizedspectral weight on pixel k of the spectrometer, and Nis the total number of energy pixels.The fine-structure descriptors that involve a deriva-tive or an energy-pixel sum (R1, Sspec) are not purelyintrinsic quantities: their absolute values depend on thespectral resolution, the energy binning, any smoothingapplied before analysis, and the experimental noise floor.R1 involves an energy derivative and so rescales un-der smoothing or spectral broadening; Sspec depends onthe effective number of spectral bins and on the noisefloor. We therefore do not treat these absolute valuesas instrument-independent, and they should not be com-pared across datasets unless the resolution and prepro-cessing are matched. In our simulations they are evalu-ated at the experimental spectral resolution.What we do compare is one level removed from theabsolute values: the spatial covariance of the descrip-tors, the inter-descriptor correlation signs, and the cor-relation lengths, all computed with the same preprocess-ing applied to the experimental and simulated spectra.These quantities are expected to be more robust to auniform rebinning or broadening, because the same op-eration acts at every spatial pixel while the pixel-to-pixelvariation is set by the underlying disorder landscape. Weverified this explicitly only for FS , whose FS–R1 corre-lation shifts by less than 0.03 under the threshold vari-ations above; we do not claim exact invariance for R1or Sspec, and a full propagation of instrumental noiseand resolution through the covariance matrix is left tofuture work. Accordingly, R1, FS , and Sspec are best re-garded as comparative descriptors defined together witha specified spectral-preprocessing protocol, rather thanas absolute instrument-independent observables; consis-tent preprocessing of experiment and simulation is whatmakes the comparison meaningful.B. Centroid Energy: Slow Disorder DominanceWe now derive the statistical properties of each de-scriptor from the disorder model.Proposition 1 (Centroid energy tracks slow disorder).When the integrated centroid is dominated by the smoothbackground envelope rather than by the integrated weightof the sharp trap lines,Ecent(R) ≈ E(0)cent + V̄s(R), (31)where E(0)cent is the disorder-free centroid energy andV̄s(R) is the optical-spot-averaged slow disorder (22).Proof. From Eqs. (17) and (24):Ecent(R) =∑nAn(R)En∑nAn(R). (32)Using the rigid background displacement En ≈ E(0)n +V̄s(Rn) from (23), and noting that An(R) is concentrated10near states ψn whose support overlaps the optical spot atR, so V̄s(Rn) ≈ V̄s(R) for all significantly contributingstates:Ecent(R) ≈∑nAn(R)(E(0)n + V̄s(R))∑nAn(R)= E(0)cent + V̄s(R).(33)Corollary 1 (Centroid correlation function).CEcent(r) ≡ ⟨δEcent(R) δEcent(R′)⟩⟨δE2cent⟩≈ exp(− r22(ξ2s + 2σ2opt)), (34)for r = |R−R′|, where δEcent ≡ Ecent − ⟨Ecent⟩.Proof. V̄s(R) =∫d2rWopt(r−R)Vslow(r) is a Gaussian-filtered version of Vslow. Setting r = |R − R′|, the con-volution of two Gaussians gives:⟨V̄s(R)V̄s(R′)⟩ =∫d2r d2r′Wopt(r−R)Wopt(r′ −R′)×W 2s e−|r−r′|2/2ξ2s≃W 2s exp(− r22(ξ2s + 2σ2opt)), (35)since the convolution of Gaussians with widths σopt, σopt,ξs yields a Gaussian with width√ξ2s + 2σ2opt. The opticalfiltering reduces the prefactor to Var(V̄s) =W 2s ξ2s/(ξ2s +2σ2opt); for ξs ≫ σopt this is ≃W 2s , as used above (equiv-alently, one may read Ws here as the filtered amplitudeWs,eff ≃ Ws). Since the normalized correlation (34) di-vides by the variance, this prefactor cancels and Eq. (34)is exact regardless of the reduction. The effective corre-lation length is:ξeff =√ξ2s + 2σ2opt. (36)Since ξs ≫ σopt (2 µm vs. 0.36 µm), we have ξeff ≈ξs.Remark 1. The optical spot smoothing is negligible whenξs ≫ σopt, which is satisfied experimentally. The mea-sured 1/e correlation length of Ecent directly yields ξs.C. Spectral Entropy and Roughness: TrapDisorder DominanceIn contrast to Ecent, the spectral entropy and rough-ness primarily reflect the local spectral manifold com-plexity generated by trap disorder.Proposition 2 (Entropy and roughness track trap den-sity). For fixed slow disorder, the spectral entropy Sspecand roughness R1 are increasing functions of the effectivetrap density nefft (R) = nt · f(V̄s(R)), where f is an in-creasing function of the slow disorder background (deeptraps attract excitons more effectively in a backgroundminimum).Justification. Each additional trap line resolvedwithin the optical spot adds total variation to I(E,R),raising R1, and adds a spectral component, raising Sspectoward its bound; both therefore increase with the num-ber of trap lines per spot, i.e. with nefft . The enhance-ment f(V̄s) grows toward slow-potential minima, wheredeeper local wells bind excitons more effectively. The ar-gument is monotonic rather than exact (near-degeneratelines contribute sub-additively), consistent with the ∝ntW2t σ2t variance scaling of Appendix C.This proposition has an important corollary: eventhe trap-sensitive descriptors Sspec and R1 should showmicron-scale spatial correlations, because the effectivetrap coupling is modulated by V̄s(R). Specifically, re-gions with lower Ecent (deeper Vslow minima) will showenhanced FS and R1 because traps in such regions createdeeper bound states relative to the local band edge.This gives the prediction:ξ(Sspec) ≈ ξ(R1) ≈ ξ(FS) ≈ ξs, (37)consistent with the experimental observation that all de-scriptors show ξ ≈ 1–2 µm.V. CORRELATION LENGTH HIERARCHYThis section derives the central analytical result ofthe paper: the correlation-length hierarchy ξ(Ecent) ≥ξ(Edom). We state it as a proposition and prove it fromthe decomposition of the dominant-peak energy into aslow part and a short-range trap-switching part. As em-phasized in the introduction, the informative content isnot the mere ordering (any average is smoother than apeak position) but its magnitude, which encodes the rel-ative strength of the trap and slow disorder.A. Main ResultProposition 3 (Correlation Length Hierarchy). For themodel (1)–(6) with Ws,Wt > 0:ξ(Ecent) ≥ ξ(Edom) ≳ ξt ≡√2σt, (38)where ξt is the trap correlation length. The first in-equality follows within the present decomposition fromthe positivity of the short-range trap-switching variance(Sec. V); the second is a microscopic floor that shouldbe read as an order-of-magnitude lower bound. In prac-tice ξ(Edom) is not set by ξt itself but by the larger trap-switching scale ℓswitch ∼ σopt and is further broadenedby optical and pixel (∆x) averaging, so the effective flooris ≳ max(ξt, σopt,∆x). The equality ξ(Ecent) = ξ(Edom)11holds when the trap-switching variance σ2δ vanishes, forexample in the limit Wt → 0 (or when the dominantpeak is always set by the smooth background, so that trapsnever switch Edom).B. ProofStep 1: Decomposition of dominant-peak energy. Thedominant-peak energy Edom is set by the position of thehighest spectral peak within the optical spot. We write:Edom(R) = V̄s(R) + δEdom(R), (39)where δEdom(R) captures the deviation of the dominantpeak from the smooth background. Unlike the centroid,which averages over all states, the dominant-peak energyis set by the single state with the highest optical weight atposition R—which is typically a trap state or a localizedmoiré state near R.Step 2: Statistics of δEdom. The fluctuations δEdomarise from: (a) switching of trap levels within the op-tical spot as R varies, (b) variation in which moiré-localized state has the highest weight. Both sources pro-duce fluctuations with correlation lengths ℓswitch ∼ σopt(scale over which trap configuration changes as the opti-cal spot moves) and ℓt ∼ σt (individual trap scale). Sinceσt ≲ σopt ≪ ξs:ξδEdom∼ σopt ≪ ξs. (40)Step 3: Correlation length of Edom. The total Edomcorrelation function is:⟨Edom(R)Edom(R′)⟩ = ⟨V̄s(R)V̄s(R′)⟩+ ⟨δEdom(R)δEdom(R′)⟩=W 2s e−r2/2ξ2eff + σ2δ h(r/σopt),(41)where r = |R−R′| (as in Corollary 1), σ2δ = Var(δEdom),and h(x) → 0 for x≫ 1. In writing Eq. (41) we have ne-glected the cross-correlation ⟨V̄s(R) δEdom(R′)⟩. Weaktrap clustering correlates the trap positions with Vslowand makes this term nonzero, but it is slaved to V̄s andtherefore only renormalizes the effective short-range vari-ance σ2δ ; it cannot make Edom smoother than Ecent, sothe ordering ξ(Ecent) > ξ(Edom) is preserved. The vari-ance satisfies:Var(Edom) =W 2s + σ2δ > W 2s = Var(Ecent), σδ > 0.(42)The normalized correlation function:CEdom(r) =W 2s e−r2/2ξ2eff + σ2δh(r/σopt)W 2s + σ2δ, (43)decays from 1 to zero at a scale smaller than ξeff becausethe σ2δh(r/σopt) term vanishes quickly while the remain-ing W 2s e−r2/2ξ2eff/(W 2s + σ2δ ) < 1 at r = 0.At the 1/e scale of the experimentally relevant hi-erarchy, ξ(Edom) ≳ 1 µm ≫ σopt, so the short-rangeterm σ2δh(r/σopt) has already decayed and only thesmooth background contribution survives. The 1/elength ξ(Edom) then satisfies the implicit equationW 2sW 2s + σ2δe−ξ(Edom)2/2ξ2s = e−1. (44)Solving Eq. (44) in the perturbative regime σ2δ/W2s < 1gives a shortened correlation length ξ(Edom) < ξs ≈ξ(Ecent); in the weak-trap-noise limit σ2δ ≪ W 2s this re-duces to the closed-form expansionξ(Edom) ≈ ξs(1− σ2δ2W 2s). (45)Outside the perturbative regime, ξ(Edom) must be ob-tained directly from Eq. (41); only the qualitative order-ing ξ(Edom) < ξ(Ecent) is preserved within the presentdecomposition.Order-of-magnitude estimate. From the experi-ment [35]: ξ(Ecent) ≈ 2.00 µm, ξ(Edom) ≈ 1.27 µm, soξ(Edom)/ξ(Ecent) ≈ 0.64. Inserting this ratio into theGaussian implicit relation (44) (with ξ(Ecent) ≈ ξs) andsolving for the trap-switching variance givesσ2δW 2s= exp[1− ξ(Edom)22ξ2s]− 1 ≈ 1.2, (46)i.e., a trap-switching fluctuation σδ ∼Ws of order unity.The independent estimate from the centroid–dominantcorrelation [Eq. (49) below, σδ/Ws ≈ 0.8] falls in thesame range; the residual spread reflects the assumedfunctional form of the mixed correlation decay, so we readthis only as an order-of-magnitude estimate. Both routesplace the trap disorder comparable to the slow disorderamplitude, in the hierarchical regime, and the simulatedcorrelation functions [Fig. 3(a)] confirm the hierarchy di-rectly, as summarized by the correlation-length bar chart[Fig. 3(b)].Range of validity. The weak-noise expansionEq. (45) is strictly controlled only for σ2δ/W2s ≪ 1.The experimentally inferred ratio σ2δ/W2s ≈ 1.2 lies out-side this perturbative regime, so the numerical inver-sion above should be read as an order-of-magnitude esti-mate based on the full two-component correlation formEq. (41), not as a small-noise expansion. The sign of thehierarchy ξ(Edom) < ξ(Ecent) is robust within the presentdecomposition for any σδ > 0, following from the positiv-ity of the trap-switching variance; only the precise map-ping σ2δ/W2s ↔ ξ(Edom)/ξs is approximation-controlled.VI. INTER-DESCRIPTOR CORRELATIONTHEORYBy an inter-descriptor correlation we mean the cor-relation between two spatial descriptor maps X(R) and120 1 2 3 4 5Distance r ( m)0.00.20.40.60.81.0Correlation C(r)1/e(a) Spatial correlation functions(Ecent) = 1.95 m(Edom) = 1.55 m(W80) = 0.61 m(FS) = 1.08 m(R1) = 1.20 m(Sspec) = 0.91 mcosine sim. = 1.85 mEcent Edom FS R1 W80 Sspec0.00.51.01.52.02.53.0Correlation length  (m)(Ecent) > (Edom)(b) Correlation length hierarchySimulation (mean±std, 30 seeds)ExperimentFIG. 3. Spatial correlation length hierarchy confirmed by simulation. (a) Normalized spatial correlation functionsC(r) for six spectral descriptors, computed from a single 20 × 20 hyperspectral map on an 8 × 8 µm2 domain with best-fitparameters (Ws = 6 meV, ξs = 2 µm, Wt = 12 meV). The horizontal dotted line marks the 1/e threshold, and the legendlists the 1/e length of each highlighted descriptor for this single realization (the ensemble means over thirty realizations areshown in panel (b)). Thin light-gray lines are the remaining, un-highlighted descriptors (∆Ecd, RHL, Itot). The dark-gray solidcurve is the whole-spectrum cosine similarity. Because every pixel shares the same broad envelope, its raw value sits on a highnonzero plateau; we therefore subtract that long-range plateau and renormalize to unity at r = 0, so that only the physicallydecaying excess is shown. Its 1/e length, 1.85 µm, matches the experimental whole-spectrum value of 1.87 µm. (b) Simulated(blue) versus experimental (red) 1/e correlation lengths. The simulated bars are the mean ± standard deviation over thirtydisorder realizations, with ξ extracted as the 1/e decay length of the spatial autocorrelation (the experimental convention). Thesimulation confirms the hierarchy ξ(Ecent) > ξ(Edom) (blue arrow): ξ(Ecent) = 1.93 ± 0.14 µm and ξ(Edom) = 1.38 ± 0.22 µm(ratio 0.72 ± 0.10), consistent with the experimental 2.00 and 1.27 (ratio 0.64) [35]. The FS and R1 lengths fall below theexperimental values, reflecting the Poisson-limited trap statistics of the simplified model (Sec. XII); their finite-size robustnessis examined in Appendix F.Y (R) evaluated over all measured pixels R of the hy-perspectral map; it thus measures how two coarse de-scriptors of the local spectrum co-vary across the sam-ple, not a correlation between individual spectral peaks.Throughout this section, correlations between two suchdescriptor fields X(R) and Y (R) are quantified by thePearson coefficientρP(X,Y ) =Cov(X,Y )√Var(X) Var(Y ), (47)where Cov(X,Y ) = ⟨XY ⟩ − ⟨X⟩⟨Y ⟩ and Var(X) =Cov(X,X), with ⟨·⟩ denoting the spatial average over themap. The experimental descriptor matrix, by contrast,is reported using the Spearman rank-correlation coeffi-cient ρS [35], defined as the Pearson correlation of therank-transformed descriptor fields,ρS(X,Y ) = ρP(rankX, rankY ), (48)where rankX denotes the rank-transformed values ofX(R) over all map pixels. It thus measures monotonic as-sociation without assuming a linear relation. The Spear-man coefficient is the natural choice here because thespectral descriptors are nonlinear functionals of the lo-cal PL spectrum and need not be Gaussian distributed;rank correlations therefore provide a robust measure ofmonotonic co-variation that is directly comparable to theexperimental descriptor-correlation matrix. We computeρP analytically because it is exact for the Gaussian dis-order variables underlying our model, and use it as asign-and-magnitude estimate for ρS: for the monotonicdescriptor relations considered here, the two coefficientsshare the same sign and have comparable magnitude, sothat the analytical ρP predicts the measured ρS. Wherewe quote our own simulation values for comparison withexperiment (e.g. Sec. XID), we report ρS to match theexperimental convention.13A. Centroid-Dominant CorrelationFrom Eqs. (31) and (39), the Pearson correlation be-tween the centroid and peak-position fields isρP(Ecent, Edom) =Cov(V̄s, V̄s + δEdom)√Var(V̄s)√Var(V̄s) + Var(δEdom)=Ws√W 2s + σ2δ< 1. (49)Here Ws denotes the optically filtered slow-disorder am-plitude Var(V̄s)1/2 (Sec. III), which is indistinguishablefrom the bare Ws in the limit ξs ≫ σopt. Identi-fying this with the experimental Spearman value [35]ρS ≈ 0.79 gives σδ/Ws ≈ 0.8, of the same order as thecorrelation-length estimate above (σδ/Ws ∼ 1); both in-dicate σδ ∼Ws.B. The ∆Ecd–RHL Anti-CorrelationThe strong experimental anti-correlationρS(∆Ecd, RHL) ≈ −0.978 [35] is the most strikinginter-descriptor correlation. We now derive this asa robust spectral shape relation for spectra with adominant unimodal envelope.Proposition 4 (∆Ecd–RHL anti-correlation). Considerspectra with a dominant unimodal envelope whose shapeis governed by a single asymmetry parameter α, with α =0 the case symmetric about the dominant peak Edom. Thecentroid-dominant offset ∆Ecd = Ecent − Edom and thehigh/low ratio RHL are then monotone in α with oppositesense: ∆Ecd and RHL − 1 vanish together at α = 0 andcarry opposite signs for α ̸= 0,∆Ecd < 0 ⇐⇒ RHL > 1,∆Ecd > 0 ⇐⇒ RHL < 1,so that ρP(∆Ecd, RHL) → −1 when α is the dominantsource of pixel-to-pixel variation. (Absent the single-parameter restriction the sign relation is a strong ten-dency rather than an identity, since ∆Ecd weights thebelow/above intensities by their mean energy distanceswhereas RHL weights them equally.)Proof. The centroid Ecent =∫EI(E)dE/∫I(E)dEweights high-energy spectral weight more heavily rela-tive to Edom when the spectrum has a high-energy tail.A low-energy tail concentrates weight below Edom, giv-ing RHL > 1 and pulling the centroid down so thatEcent < Edom and ∆Ecd < 0. Conversely, a high-energytail gives RHL < 1 and ∆Ecd > 0.More precisely, define m>1 =∫∞EdomEI(E)dE andm<1 =∫ Edom−∞ EI(E)dE. Then:Ecent − Edom =m>1 +m<1Itot− Edom=m>1 − EdomI>totItot+m<1 − EdomI<totItot,(50)where I>/<tot are the integrated intensities above/belowEdom. Both terms represent signed “moments” of spec-tral weight about Edom, which are monotone functions ofspectral asymmetry.The ratio RHL = I<tot/I>tot is a monotone function ofthe same asymmetry parameter α. Since both ∆Ecd andRHL − 1 are monotone in α and vanish at α = 0 withopposite sign, their joint locus is a monotone curve ofnegative slope, giving ρP(∆Ecd, RHL) → −1 in the limitof pure asymmetry variation. This argument refers tothe asymmetry of the broad envelope about its domi-nant maximum, not to the placement of individual lines:an isolated low-energy trap peak can itself set Edom andproduce ∆Ecd > 0 (as in the trap-rich pixels of Fig. 6),but the observed near-one-dimensional ∆Ecd–RHL rela-tion indicates that it is the envelope asymmetry, ratherthan such discrete level switching, that remains the con-trolling variable across the map. Because both descrip-tors are referenced to the dominant peak, the anticorrela-tion requires this shared reference: in our simulated mapsρS(∆Ecd, RHL) = −0.97 ± 0.01 with the Edom split, butonly +0.50 with an Ecent split and +0.11 with a fixed-median split—showing that the strong relation is tiedto the dominant-peak-referenced definition (the experi-mentally adopted one), rather than a consequence of anarbitrary normalization choice.Remark 2. The near-perfect anti-correlation ρS ≈−0.978 observed experimentally indicates that spectralasymmetry variation—rather than peak multiplicity orpeak energy modulation—is the dominant source of de-scriptor variation in this dataset. This is a nontrivial di-agnostic: the mere observation that ρS(∆Ecd, RHL) ≈ −1is difficult to reconcile with simple models in which mul-tiple incoherent peaks at unrelated energies dominate thespectrum, and instead establishes that the broad enveloperemains effectively unimodal across the map—directlyconfirming the two-fluid picture of Sec. VIII.C. Sharp Fraction – Roughness CorrelationBoth FS and R1 measure the density of sharp spectralstructure. They differ in how the structure is weighted:FS measures the fraction of integrated intensity in nar-row features, while R1 measures the total variation of thespectral profile. For a spectrum consisting of a smoothbackground plus sharp peaks, both increase with thenumber and depth of sharp peaks, giving a large posi-tive correlation.14Analytically, for a spectrum I = Ibg+∑iAigη(E−Ei):R1 ≈ Rbg1 +2η√2π·∑iAiItot, (51)FS ≈∑iAiItot, (52)giving R1 ≈ Rbg1 + const× FS , and thus ρP(FS , R1) → 1when trap peak variation dominates. The experimentalvalue ρS ≈ 0.901 [35] confirms this.D. Width – Entropy CorrelationThe quantile width W80 and spectral entropy Sspecboth measure the spread of spectral weight. For a spec-trum spanning Neff effective resolution elements: W80 ∝Neff and Sspec ∝ lnNeff . Since both are monotone func-tions of Neff , they are positively correlated. The exper-imental value ρS ≈ 0.846 [35] is consistent. The fullsimulated Spearman inter-descriptor matrix is shown inFig. 4.VII. DISORDER PARAMETER REGIMESThe qualitative behavior of the model is set by the twodisorder strengths—the slow-disorder amplitude Ws andthe effective trap disorder Γt—measured against the ho-mogeneous linewidth η. Which of these exceeds η deter-mines whether the local spectra are essentially feature-less, smoothly modulated on the micron scale, denselypeaked, or organized on both scales at once. In this sec-tion we map that dependence onto four qualitatively dis-tinct regimes, in the spirit of a phase diagram (Fig. 5),and use the map to locate the experimental system. Thismatters because the descriptor correlation-length hier-archy studied here is prominent only when both disor-der scales exceed η; identifying the regime therefore tellsus when the descriptor analysis applies. We first intro-duce the two control parameters, then define the regimeboundaries, and finally place the MoSe2/WSe2 system onthe resulting map.A. Control ParametersWe map the parameter space of the two disorderstrengths into qualitatively distinct regimes:Ws (slow disorder amplitude, meV),Γt ≡ ntW2t σ2t (effective trap disorder, meV2). (53)The quantity Γt is the effective trap-potential variance,combining the trap density nt, the depth variance W 2t ,and the trap area ∼ σ2t through the dimensionless cover-age ntσ2t ; its square root Γ1/2t is the rms trap-potentialamplitude (in meV), directly comparable to Ws and η.B. Regime BoundariesWe define four regimes based on the hierarchy ofspectral disorder. The boundaries below are order-of-magnitude criteria intended to demarcate qualitativelydistinct regimes; the exact boundaries depend on mi-croscopic details and should be regarded as schematic.The geometric factors (aM/ξs)1/2 that appear beloware schematic coarse-graining factors estimating hownanoscale trap fluctuations (on the moiré scale aM ) sur-vive averaging over a slow-disorder domain of size ξs; theyare heuristic, not the result of a microscopic derivation.a. Regime I: Uniform Excitonic (small Ws, smallΓt). Neither disorder scale produces significant spectralstructure. The PL is a single smooth Gaussian-like peakat the unperturbed exciton energy. Boundary: Ws < η(spectral resolution) and Γ1/2t < η.b. Regime II: Smooth Correlated-Domain (large Ws,small Γt). Slow disorder creates micron-scale spectraldomains. The spectrum at each pixel is smooth (lowroughness, low entropy, small FS). Ecent maps showsmooth spatial modulation with ξ(Ecent) ≈ ξs. Bound-ary: Ws > η and Γ1/2t < Ws · (aM/ξs)1/2.c. Regime III: Random Line-Rich (small Ws, largeΓt). Dense sharp spectral peaks appear at random po-sitions. High roughness and entropy, but no spatial do-main structure. Boundary: Ws < η and Γ1/2t > η.d. Regime IV: Hierarchical Disorder-Dominated(large Ws, large Γt). Both disorder scales are active.Micron-scale spectral domains coexist with local mul-tipeak manifolds. The descriptor correlation-lengthhierarchy ξ(Ecent) > ξ(Edom) is maximally observableand directly measurable in this regime.Regime IV: Ws > η,Γ1/2t > Ws · (aM/ξs)1/2. (54)C. Location of the Experimental SystemThe experimental data directly constrain the modelparameters. The measured ξ(Ecent) ≈ 2.00 µm impliesξs ≈ 2 µm; the centroid energy variation across the mapimplies Ws ∼ 10–15 meV; and the prevalence of multi-peak spectra with FS > 0.1 and R1 ∼ 0.3–0.5 meV−1confirms Γ1/2t > η ∼ 2 meV. The experimental systemtherefore falls within Regime IV (hierarchical disorder-dominated), as illustrated in the disorder parameter map(Fig. 5). The value Ws ∼ 10–15 meV here is the rawcentroid variation across the experimental map. In thenumerical simulations of Sec. XI we use Ws = 6 meVas an effective slow-disorder amplitude that accounts foroptical-spot averaging and finite-map normalization, sothat the two values refer to different operational defini-tions of the slow-disorder strength rather than a quanti-tative inconsistency.15I tot E centE dom E cd W 80 R HL F S R 1S specItotEcentEdomEcdW80RHLFSR1Sspec1.00 -0.53 -0.31 -0.04 -0.39 0.14 0.60 0.61 -0.59-0.53 1.00 0.77 -0.18 0.39 0.06 -0.74 -0.79 0.64-0.31 0.77 1.00 -0.69 0.29 0.62 -0.55 -0.61 0.45-0.04 -0.18 -0.69 1.00 -0.14 -0.96 0.07 0.15 -0.10-0.39 0.39 0.29 -0.14 1.00 0.02 -0.55 -0.56 0.850.14 0.06 0.62 -0.96 0.02 1.00 0.05 -0.03 -0.030.60 -0.74 -0.55 0.07 -0.55 0.05 1.00 0.91 -0.800.61 -0.79 -0.61 0.15 -0.56 -0.03 0.91 1.00 -0.84-0.59 0.64 0.45 -0.10 0.85 -0.03 -0.80 -0.84 1.00Descriptor Spearman correlation matrix(gold box: reported experimental pairs)1.000.750.500.250.000.250.500.751.00Spearman FIG. 4. Spearman inter-descriptor correlation matrix from simulation. Each cell shows the Spearman rank correlationcoefficient ρS between a pair of spectral descriptors, computed from a simulated 20 × 20 hyperspectral map with best-fitparameters. Gold boxes mark the four pairs also reported experimentally [35]—∆Ecd–RHL, FS–R1, W80–Sspec, and Ecent–Edom. The displayed coefficients are those of a single best-fit map; the five-seed mean simulated values and their measuredcounterparts are compared in Table II (mean absolute deviation 0.010 per pair). Note the strong positive block structure among(Itot, FS , R1) (trap-rich pixels) and the negative block coupling these to (Ecent, Edom), reflecting the low-energy trap emissionat potential minima.VIII. TWO-FLUID LDOS STRUCTUREThe eigenstates of H = −(ℏ2/2M)∇2 + Vslow + Vtrapfall into two populations based on their energy relativeto the local slow potential.Background states (En ≳ V̄s(Rn)) are weaklybound to slow-disorder minima, with spatial extentsℓn ≫ σt. Each contributes a broad Lorentzian ofwidth ηbg centered near V̄s(Rn). The incoherent sum of∼20 background eigenstates per optical spot produces asmooth, quasi-Gaussian envelope that tracks V̄s(R) andconstitutes the broad background Ibg.Trap states (those lying more than ∼Wt below thelocal background, En ≲ V̄s(Rn)−Wt) are spatially con-centrated near individual trap sites Ri, with spatial ex-tent ℓn ∼ σt. Each contributes a narrow peak of widthη ≪ ηbg at energy En ≈ V̄s(Ri) + E(trap)n .The resulting local density of states at position R is:ρ(E,R) ≈ ρbg(E − V̄s(R))︸ ︷︷ ︸smooth background+∑i:Ri∈spot(R)ρtrapi (E − V̄s(Ri))︸ ︷︷ ︸sharp trap peaks, (55)and the local PL spectrum follows from integratingρ(E,R) against the optical weight function. This two-fluid structure is reproduced at the model level by theHamiltonian diagonalization in Fig. 8(b): the filled areasshow the background and trap contributions separately.A key non-trivial consequence is that the broad-background intensity Ibg,tot and sharp-peak intensityIsharp,tot are positively correlated:Cov(Ibg,tot, Isharp,tot) > 0. (56)This follows because both populations are enhanced inslow-potential minima: background eigenstates are pref-erentially concentrated in slow-potential minima by dis-order selection, while traps are preferentially nucleated atthe same sites. The positive covariance distinguishes thehierarchical disorder landscape from a two-phase compe-tition model, in which sharp peaks would form at theexpense of the background.IX. SPECTRAL FAMILIES AS CORRELATEDDISORDER DOMAINSA. Origin of Spectral FamiliesThe experimental observation of three dominant spec-tral families in PCA + Gaussian mixture clustering raises16100 101Slow disorder Ws (meV)100101Trap disorder Wt (meV)Ws=Wt =IweakdisorderIIslow-dominatedIIItrap-dominatedIVhierarchicaldisorderDisorder parameter regimesBest-fit (Ws = 6, Wt = 12 meV)FIG. 5. Disorder parameter map for the hierarchi-cal disorder model. Four qualitatively distinct regimes aredefined by the disorder strengths relative to the single-levellinewidth η = 1.5 meV: Regime I (weak disorder, Ws,Wt <η): spectrally uniform emission; Regime II (slow-dominated,Ws > η > Wt): smooth micron-scale spectral modulation;Regime III (trap-dominated, Wt > η > Ws): dense randomsharp lines without spatial organization; Regime IV (hierar-chical disorder-dominated, Ws,Wt > η): coexisting micron-scale domains and dense local trap manifolds; the descrip-tor correlation-length hierarchy is maximally visible. Thegold star marks the best-fit parameters reproducing the fourprincipal Spearman descriptor correlations reported experi-mentally [35] simultaneously. Here nt and σt are held fixed,so the vertical axis Wt represents the trap-disorder strengthΓ1/2t = (ntW2t σ2t )1/2 of Eq. (53) up to a constant factor.the question: are these families distinct thermodynamicphases, or emergent statistical features of the disorderlandscape?In our framework, spectral families emerge naturallyfrom the correlated slow disorder field Vslow(r). AGaussian random field on a finite domain [0, L]2 withcorrelation length ξs ≪ L develops Ndom ∼ (L/ξs)2quasi-independent regions with characteristic local val-ues Vslow ∼Ws.For the experiment [35]: L ≈ 8 µm, ξs ≈ 2 µm, givingNdom ∼ 16 statistically independent regions. However,PCA+clustering of a 9-dimensional descriptor space mayresolve only the dominant modes of variation, yielding∼ 3 effective families.Remark 3 (Spectral family number (heuristic esti-mate)). We expect the number of spectral families re-solved by Gaussian mixture clustering on the first k prin-cipal components of the descriptor space to be Nfam ≈min(k + 1, Ndom) in the regime where inter-domain de-scriptor separation exceeds intra-domain variance. Thek+1 form is a heuristic counting estimate (each retainedmode adds at most one resolvable cluster boundary), nota sharp theorem.B. Statistical Nature of Spectral FamiliesThe spectral families in the present framework aretherefore not sharply distinct thermodynamic phases(like phase-separated domains with a definite interfa-cial free energy). Instead, they are emergent statisticalclusters of the continuously varying slow disorder land-scape, identified by a finite number of PCA modes andseparated by soft boundaries that represent continuouscrossovers rather than sharp interfaces.This interpretation carries several testable conse-quences. The cluster boundaries should be smooth, andthe number of identifiable families should grow with thePCA truncation order rather than saturating at a ther-modynamically fixed count. Different spectral descrip-tors should produce consistent but not identical familyassignments, since each descriptor filters a different com-bination of disorder scales. The characteristic domainsize of each family should equal ∼ ξs, in agreement withthe experimentally observed mean domain diameter of1.79 µm [35].X. MULTI-SCALE SPECTRAL ORGANIZATIONIN THE HIERARCHICAL REGIMEThe hierarchically disordered regime (Regime IV, bothWs ≫ η and Γt ≫ η2) exhibits multi-scale spectral orga-nization that is absent in any single-disorder model. Theeffective potential Veff = Vslow + Vtrap supports a nestedlandscape: the smooth background set by Vslow organizesthe micron-scale spectral envelope, while Vtrap providesthe fine structure within each optical-spot region.A quantitative diagnostic for the degree of multi-scale organization is the mean spectral cosine similar-ity between spatially separated pixels. For the mini-mal model, the spatial correlation of the full spectraCI(r) ≡ ⟨I(E,R)·I(E,R+r)⟩/⟨∥I∥2⟩ decays on a lengthscale set by ξs. The baseline-subtracted cosine simi-larity of our simulated map (Fig. 3) decays with a 1/elength of 1.85 µm, in close agreement with the whole-spectrum cosine correlation length of 1.87 µm reportedexperimentally [35]. The same landscape that producesthe descriptor-specific correlation lengths (Sec. V) alsogoverns the domain structure visible in PCA cluster maps(Sec. IX).The key observable consequence is that the spectral co-variance is not self-averaging on the experimental map:different 8×8 µm realizations produce statistically similardescriptor distributions but different spatial patterns, re-flecting the finite ratio L/ξs ≈ 4 of map size to correlationlength. This non-self-averaging is a direct signature ofthe multi-scale organization and distinguishes the hierar-17chical regime from both the smooth-domain (Regime II)and trap-dominated (Regime III) limits.XI. NUMERICAL SIMULATIONSA. Overview: Two Complementary ApproachesNumerical validation of the analytical predictions pro-ceeds via two complementary routes on a common 128×128 grid spanning an 8 µm ×8 µm domain (∆x =62.5 nm).Route 1 (primary): Hamiltonian diagonaliza-tion. The Hamiltonian acts as a coarse-grained spec-tral generator : it defines a spatially correlated eigenvaluespectrum whose density of states, when convolved withthe optical point-spread function, produces a well-posedLDOS at every map position. The key property is thatthe spectral descriptors at a given position depend onwhich eigenstates are in competition for spectral weightwithin the optical spot—a many-eigenstate quantity thatrequires solving the full eigenvalue problem rather thanreading off local potential values.Concretely, the full sparse exciton Hamiltonian H =−(ℏ2/2M)∇2 + Vslow + Vtrap is assembled and diago-nalized for its lowest optically active eigenstates viathe Lanczos algorithm (ARPACK) [64]; for larger sys-tems the same low-lying spectrum can be obtained moreeconomically with shift-and-invert sparse eigensolverssuch as SLEPc/PETSc [65, 66], allowing a finer spa-tial grid. The diagonalization yields eigenstates (εn, ψn)from which the local PL spectrum is constructed asI(E,Rj) =∑nAjn gσn(E − εn) with the incoherentlocal-density optical weights Ajn [Eq. (16)]; Appendix Econfirms that the coherent weight (13) yields statisti-cally indistinguishable descriptor statistics within numer-ical uncertainty. This route verifies that the correlation-length hierarchy and inter-descriptor Spearman relationsemerge from eigenvalue statistics without relying onthe synthetic two-fluid spectrum construction of Route2. In the limit of vanishing hopping (kinetic) energythop ≡ ℏ2/(2M∆x2) → 0 (the finite-difference hoppingof Appendix B), the model reduces to a spatially corre-lated random spectral manifold with no quantum trans-port; the finite thop acts solely as a spectral regularizerthat converts the discrete disorder landscape into a con-tinuous, normalizable LDOS. Results are presented inSec. XI F.Route 2 (computational): PhenomenologicalPL model. As a computationally efficient alternativeenabling large-ensemble parameter sweeps, we directlysynthesize local PL spectra from the two-fluid physicalpicture as:I(E,R) = Ibg(E; V̄s(R))+ ftrap∑k∈N (R)wk(R) gη(E − Ek), (57)where V̄s(R) =∫Wopt(r − R)Vslow(r) d2r is the op-tically averaged slow potential, Ibg is a Gaussian ofwidth ηbg = 10η, wk(R) = exp(−|Rk − R|2/2σ2opt)is the optical weight at trap k, Ek = V̄s(Rk) + ukis the trap energy, and ftrap = 0.2 is chosen so thatFS ∼ 0.15–0.40. The model parameters ftrap and ηbgare not independently derived from the Hamiltonian;they are calibrated to the experimental sharp-fractionrange. We emphasize that ftrap is not used to fit thedescriptor correlation signs or magnitudes: it only setsthe overall sharp-fraction scale ⟨FS⟩ and enters the in-verse parameter extraction only through that calibra-tion, while the main correlation signs and relative magni-tudes are controlled primarily by the disorder parameters{Ws, ξs,Wt, nt, Ebias}. This route enables multi-seed pa-rameter sweeps for the Spearman-correlation comparison(Sec. XID); the correlation-length statistics in Fig. 3 andAppendix F are evaluated over thirty independent disor-der realizations.B. Trap Clustering ModelIf trap positions are drawn from an uncorrelated Pois-son process, then ξ(FS) ∼ σopt ≪ ξs. Experimentally,however, ξ(FS) ≈ ξ(Ecent) ≈ ξs, suggesting that the trapdensity tracks the slow disorder landscape. This is physi-cally motivated by the possibility that reconstruction do-mains, stacking-registry variations, strain concentrators,or charged defects correlate the locations, or the opticalactivation, of trap-like emitters with the slow exciton-energy landscape.We model this by the minimal correlated-trap ansatz :trap positions are drawn fromP (r) ∝ exp(−Vslow(r)Ebias), (58)where Ebias > 0 is a trap–slow-disorder bias energy scale(not a temperature) that sets the degree of spatial corre-lation between optically active traps and slow-potentialminima. This is the lowest-order ansatz that capturesthe coupling between the two disorder scales without in-troducing additional clustering parameters beyond Ebias;the microscopic processes behind the correlation (defectmigration, strain coupling) are not modeled explicitly. Inthe limit Ebias → ∞, the distribution is uniform (randomtraps). For Ebias ∼Ws, over typical one-sigma variationsof Vslow the trap density changes by a factor of ordereWs/Ebias ; rare ±3σ excursions (∆V ∼ 3Ws) can producea much larger contrast, n(r)/⟨n⟩ ∼ e±3. This createsdomains of high and low trap density with correlationlength ξs.The spatial correlation of the trap density field is:Cn(r) ≡ Corr[n(R), n(R+ r)]=exp[(Ws/Ebias)2 Cs(r)]− 1exp[(Ws/Ebias)2]− 1, (59)18which inherits a correlation scale of order ξs for finitebias strength. This expression follows from the moment-generating function of the Gaussian field Vslow applied tothe log-normal trap-density field n(r) ∝ e−Vslow(r)/Ebias :for jointly Gaussian Vslow(R), Vslow(R + r) with co-variance W 2s Cs(r), one has ⟨n(R)n(R + r)⟩/⟨n⟩2 =exp[(Ws/Ebias)2 Cs(r)], which normalizes to the correla-tion coefficient above. The observable FS inherits thiscorrelation length when the clustering signal exceeds thePoisson noise in trap count per optical spot.C. Parameter OptimizationThe model has five parameters: {Ws, ξs,Wt, nt, Ebias}(with σt = 50 nm and η = 1.5 meV fixed). We deter-mine these by minimizing the total deviation from thefour experimentally reported Spearman inter-descriptorcorrelations:L(θ) =∑(a,b)|ρsim(a, b)− ρexp(a, b)| , (60)where the sum runs over the four pairs: (∆Ecd, RHL),(FS , R1), (W80, Sspec), (Ecent, Edom). This objective di-rectly tests the theoretical predictions of Sec. VI.The best-fit parameters (averaged over five indepen-dent disorder realizations) are:Ws = 6.0 meV, ξs = 2.0 µm, Wt = 12.0 meV,nt = 2.0 µm−2, Ebias = 7.0 meV. (61)The ratio Ws/Wt = 0.5 places the system in the trap-dominated energy-spread regime (Regime IV of Sec. VII),consistent with the experimental observation of densemultipeak spectra. The slow disorder amplitude Ws =6.0 meV sets the centroid energy variation scale, whilethe trap depthWt = 12.0 meV determines the dominant-energy variation and the Ecent–Edom decorrelation ratio.The low-trap-fraction limit (ftrap = 0.2 ≪ 1) pre-dicts ρP(Ecent, Edom) → Ws/√W 2s + W̄ 2t where W̄ 2t =W 2t (1 − 2/π) ≈ 0.363W 2t for the half-normal distribu-tion. With Ws = 6, Wt = 12: theoretical ρP,min =6/√36 + 52.3 = 0.638, which rises to ≈ 0.79 at finiteftrap = 0.2 through the partial E-dom–E-bg alignment,matching the observed value.D. Key Numerical ResultsFigure 6 summarizes the representative PL spectraand spectral descriptor maps from the best-fit simula-tion: three representative pixel spectra [Fig. 6(a)–(c)],the ∆Ecd–RHL scatter plot [Fig. 6(d)], and the spatialmaps of δEcent, δEdom, FS , and R1 [Fig. 6(e)–(h)].With parameters (61), the five-seed mean Spearmancorrelations are (the Spearman sweep uses five seedsfor the parameter scan, whereas the correlation-lengthstatistics in Fig. 3 and Appendix F use thirty indepen-dent disorder realizations):Within the phenomenological PL model, all four cor-relations are reproduced to within the seed-to-seed stan-dard deviation, confirming the theoretical predictions ofSec. VI. The ∆Ecd–RHL anti-correlation is reproduced towithin 0.002 (ρsim = −0.976± 0.009 vs. ρexp = −0.978),consistent with the robust spectral shape relation ofProposition 4. The FS–R1 correlation is equally wellmatched (ρsim = +0.903 ± 0.024 vs. +0.901); the smallseed-to-seed variability reflects the discrete nature of trapcounting within each optical spot. The W80–Sspec corre-lation is slightly underestimated (ρsim = +0.821± 0.065vs. +0.846), a discrepancy attributable to the absence ofthe moiré potential, which would contribute additionalspectral structure and broaden W80. Finally, the Ecent–Edom correlation, the most sensitive diagnostic of thetrap-to-slow disorder ratio, is reproduced within the seed-to-seed uncertainty (ρsim = +0.800± 0.075 vs. +0.788).The simulated 20×20 hyperspectral map also confirmsthe correlation length hierarchy (Proposition 3):ξ(Ecent) = 1.93 ± 0.14 µm > ξ(Edom) = 1.38 ± 0.22 µm(mean ± standard deviation over thirty disorder re-alizations, using the 1/e decay length of the spatialautocorrelation—estimated from the empirical semivar-iogram [67]—to match the experimental convention),with ratio ξ(Edom)/ξ(Ecent) = 0.72 ± 0.10 (experiment:1.27/2.00 = 0.64 [35]). The corresponding standard er-rors of the ensemble means are approximately 0.03 µmfor Ecent and 0.04 µm for Edom, much smaller than the≈ 0.55 µm separation between the two mean correlationlengths, so the ordering is well resolved by the ensem-ble. The centroid correlation length now agrees with theexperimental 2.00 µm; the finite-size robustness of thehierarchy—and of the ratio—is examined in Appendix F.The simulated correlation lengths ξ(FS) = 0.84 ±0.23 µm and ξ(R1) = 0.93 ± 0.20 µm remain below theexperimental values of ∼ 2 µm. This discrepancy is aknown limitation of the simplified random-trap model(see Sec. XII), where with nt = 2 µm−2 there are only∼2–3 traps per optical spot, causing Poisson noise to par-tially mask the spatial trap-density correlation. The realmoiré system contains ∼ 103–104 moiré-scale sites peroptical spot, and even a small optically active fractionmay be spatially organized by reconstruction, strain, orstacking-registry variations. Such correlations provide anatural route toward ξ(FS) ∼ ξs, beyond the Poisson-limited random-trap model.Details of the numerical implementation are providedin the analysis scripts available upon reasonable request(see the Data Availability statement).1950 25 0E (meV)00.05I (a.u.)(a) background-dominatedEcent=1.2Edom=1.350 25 0E (meV)(b) mixedEcent=-1.9Edom=-0.850 25 0 25E (meV)(c) trap-richEcent=-15.5Edom=-10.90 10 20Ecd (meV)012R HLEcd RHL(d) S = 0.9550.00.20.40.6F S0 2 4 6 8x ( m)02468y (m)(e) Ecent (meV)0 2 4 6 8x ( m)(f) Edom (meV)0 2 4 6 8x ( m)(g) FS0 2 4 6 8x ( m)(h) R1 (meV 1)100100.00.20.40.60.100.150.20FIG. 6. Simulated local PL spectra and spectral descriptor maps. Upper row: three representative PL spectra fromthe 20×20 measurement map, illustrating the three spectral types present in the hierarchical disorder regime. (a) Background-dominated pixel: a single broad Gaussian background peak with Ecent ≈ Edom and no resolved sharp lines (FS ≈ 0). (b) Mixedpixel: the background peak is visible alongside one or two sharp trap peaks; the centroid Ecent is pulled below the dominantpeak Edom by the trap-related weight on the low-energy side of the envelope. (c) Trap-rich pixel: several sharp lines above thebackground, with the centroid pulled toward the cluster of trap energies. Vertical dashed and dotted lines mark Ecent (red)and Edom (blue) for each pixel. (d) Scatter plot of ∆Ecd vs. RHL for all 400 pixels, colored by FS . The tight anti-correlation(ρS = −0.955 for this single representative seed; cf. the five-seed mean ρS = −0.976 ± 0.009 in Table II) reflects the robustspectral shape relation derived in Proposition 4; pixels with larger FS (yellow) show the strongest separation between centroidand dominant-peak energy. Lower row: spatial maps of four spectral descriptors across the 20 × 20 measurement grid. (e)δEcent = Ecent−⟨Ecent⟩: smooth micron-scale fluctuations tracking Vslow, with ξ(Ecent) ≈ 1.93 µm. (f) δEdom = Edom−⟨Edom⟩:more fragmented pattern with additional trap-switching fluctuations, confirming ξ(Edom) < ξ(Ecent). Panels (e) and (f) sharea common color scale (in meV), so the larger fluctuation amplitude of δEdom is apparent. (g) FS map: trap-density-modulatedsharp-fraction distribution. (h) R1 (spectral roughness) map: closely tracks FS , consistent with ρS(FS , R1) = 0.909.Descriptor pair Simulation (mean ± std) Experiment(∆Ecd, RHL) −0.976 ± 0.009 −0.978(FS , R1) +0.903 ± 0.024 +0.901(W80, Sspec) +0.821 ± 0.065 +0.846(Ecent, Edom) +0.800 ± 0.075 +0.788TABLE II. Spearman inter-descriptor correlations: simulation vs. experiment. Simulation values are means over five indepen-dent disorder realizations; the experimental values are from Ref. [35]. The total score L = 0.041 represents a mean absolutedeviation of 0.010 per pair.E. Necessity of Hierarchical Disorder: Single-ScaleComparisonA natural question is whether either disorder compo-nent alone could reproduce the observed phenomenology.To address this we repeat the phenomenological simula-tion in three settings, keeping all other parameters fixedat the best-fit values: (i) slow only (Wt = 0, no traps);(ii) trap only (Ws = 0, no slow background); (iii) hierar-chical (both, as above). The results are summarized inTable III and Fig. 7, which contrasts the correlation func-tions of the slow-only [Fig. 7(a)], trap-only [Fig. 7(b)],20and hierarchical [Fig. 7(c)] models.The slow-only model fails because Ecent and Edom be-come statistically identical: with no traps, the spectrumat every pixel is a single broad envelope centered on V̄s(r),so ρS(Ecent, Edom) = +1.00 and ξ(Ecent) = ξ(Edom) =1.98 µm. No correlation-length hierarchy is present, and⟨FS⟩ = 0 (no sharp peaks).The trap-only model fails in the opposite way: in theabsence of a smooth background, Ecent and Edom bothinherit only the spatial scale of the optical spot, giv-ing ξ(Ecent) = 0.23 µm and ξ(Edom) = 0.18 µm, bothwell below the experimental ∼ 1.27–2.00 µm [35] and be-low the optical spot size σopt. Pixels develop sharp lines(⟨FS⟩ = 0.34) and a strong ρS(∆Ecd, RHL) = −0.98, butno micron-scale descriptor domains appear, in contradic-tion with the experiment.Only the hierarchical model simultaneously reproduces(i) the correlation-length ordering ξ(Ecent) > ξ(Edom)at the micron scale, (ii) the strong ∆Ecd–RHL anti-correlation, and (iii) the observed sharp-fraction values.This is the most direct statement of the need for two dis-order scales: descriptor covariance alone argues againsteither single-scale model from a hyperspectral PL map.F. Continuum LDOS Validation of the DescriptorHierarchyTo confirm that the correlation-length hierarchy andthe principal Spearman relations emerge from eigenvaluestatistics rather than from the construction of the phe-nomenological PL model, we repeated the analysis usingthe full sparse exciton Hamiltonian (1) diagonalized nu-merically.a. Setup. We build the kinetic-energy operator as afinite-difference Laplacian (Appendix B) on the 128×128grid and add Veff = Vslow + Vtrap on the diagonal, ob-taining a 16 384× 16 384 sparse matrix. The 1500 lowesteigenstates (εn, ψn) are computed with the Lanczos algo-rithm (ARPACK) [64] using the same disorder realizationand best-fit parameters (61).With M = 1.2m0 and ∆x = 62.5 nm, the hop-ping energy is thop = ℏ2/(2M∆x2) = 0.0081 meV,four orders of magnitude smaller than Ws = 6 meVand Wt = 12 meV. The system is therefore in thedisorder-dominated spectral-statistics regime (thop/Wt ≈7 × 10−4): the disorder potential far exceeds the ki-netic energy at the grid scale, so the eigenstate energystatistics are determined primarily by the disorder land-scape. The Hamiltonian therefore serves as a coarse-grained spectral generator : the kinetic term defines thecontinuum LDOS manifold, while the disorder landscapecontrols the spatial arrangement of eigenstates at scalesrelevant to the optical spot. Solving the eigenvalue prob-lem is then essential because the spectral descriptors at agiven position are determined by the competition amongmany eigenstates whose weight falls inside the opticalspot—a many-eigenstate quantity that cannot be readoff from local potential values. Because the IPR is dom-inated by disorder rather than quantum interference inthis regime, we use an energy-based classification instead:state n is trap-like if εn < Vslow(Rn)− 0.7Wt (i.e., lyingmore than 8.4 meV below the local background), whereRn = argmaxr |ψn(r)|2 is the localization center. Of the1500 lowest states, 126 are classified as trap-like, close tothe 133 trap positions in the disorder realization. Back-ground states receive linewidth ηbg = 15 meV; trap statesreceive η = 1.5 meV (same values as the phenomenolog-ical model).b. Results. The optical-weight matrix Ajn =∑rWopt(r−Rj)|ψn(r)|2 is computed by a single matrixmultiply, and PL spectra are synthesized asI(E,Rj) =∑nAjn gσn(E − εn), (62)where σn = η/√8 ln 2 or σn = ηbg/√8 ln 2 depending onstate type. The resulting Spearman correlations are:ρS(∆Ecd, RHL) = −0.971 [exp: −0.978],ρS(FS , R1) = +0.941 [exp: +0.901],ρS(Ecent, Edom) = +0.496 [exp: +0.788].The first two correlations agree with experiment towithin 0.04, confirming that the ∆Ecd–RHL anti-correlation and the FS–R1 co-variation emerge fromeigenvalue statistics without relying on the synthetictwo-fluid spectrum construction. The correlation-lengthhierarchy is also reproduced: ξ(Ecent) = 1.18 µm >ξ(Edom) = 0.69 µm, with ratio 0.58 (experiment: 0.64).The ρS(Ecent, Edom) value is underestimated (+0.496vs. +0.788). This is a structural consequence of the trap-dominated spectral regime: the narrow trap-state peaks(σ = 0.64 meV) are ∼10× taller than the broad back-ground peaks (σ = 6.4 meV) for the same optical weight,so Edom is almost always set by the deepest trap in theoptical spot rather than by the smooth background. Thephenomenological model avoids this artefact by assign-ing the background peak amplitude independently of thetrap amplitude (via ftrap = 0.2). Physically, the dis-crepancy reflects the fact that in the real material theradiative linewidths of individual eigenstates depend ontheir spatial character (phonon-bath coupling, oscillatorstrength) in ways that go beyond the present disorder-statistics framework.Taken together, the two routes bracket the real sys-tem: the Hamiltonian diagonalization operates in thedisorder-dominated limit (eigenstates determined purelyby the potential landscape, uniform radiative coupling),while the phenomenological model operates in the ef-fective optical-weight limit (trap intensities assigned in-dependently of eigenstate character). The experimentalsystem lies between these limits, and the fact that bothroutes reproduce the ∆Ecd–RHL and FS–R1 spectral-shape correlations to within 0.04 confirms that thesecorrelations are robust features of the disorder hierar-chy rather than artefacts of either modeling choice (the21TABLE III. Comparison of single-scale and hierarchical disorder models. Numerical values are computed from the same20× 20 hyperspectral simulation pipeline with identical Ws, Wt, ξs, nt, and η as in the rest of this section, varying only whichcomponents are switched on. Experimental values are from Ref. [35].Model ξ(Ecent) ξ(Edom) ρS(∆Ecd, RHL) ⟨FS⟩(µm) (µm)Slow only (Wt = 0) 1.98 1.98 −0.47 0.00Trap only (Ws = 0)b 0.23 0.18 −0.98 0.34Hierarchical 1.93 1.38 −0.97 0.30Experimenta 2.00 1.27 −0.978 0.15–0.40a Ref. [35].b Correlation lengths are the 1/e decay length of the spatial autocorrelation (matching the experimental convention), averaged overthirty disorder realizations. For the trap-only model the descriptor fields are short-range and near-degenerate, so their autocorrelationdoes not reach 1/e within the map; the quoted trap-only lengths use a Gaussian fit C(r) = e−r2/2ξ2 , which resolves the short trapscale.0 1 2 3 4r ( m)0.00.20.40.60.81.0C(r)(a) Slow onlyEcentEdom0 1 2 3 4r ( m)(b) Trap onlyEcentEdom0 1 2 3 4r ( m)(c) HierarchicalEcentEdomFIG. 7. Single-scale disorder cannot reproduce the correlation-length hierarchy. Spatial correlation functionsC(r) of Ecent (blue circles) and Edom (red squares) for the three disorder models, shown for one representative realization;the corresponding ensemble 1/e correlation lengths are listed in Table III. (a) Slow only: the two descriptors are statisticallydegenerate; their correlation functions are indistinguishable, giving no hierarchy. (b) Trap only: both decay within the opticalspot (ξ < σopt); no micron-scale descriptor domains form. (c) Hierarchical disorder: ξ(Ecent) > ξ(Edom), reproducing theexperimentally observed ordering. Horizontal dotted line marks C(r) = 1/e.Ecent–Edom value, discussed above, is the one correla-tion that is regime-sensitive). For the same reason,the two routes are not expected to yield identical ab-solute correlation lengths (phenomenological: ξ(Ecent) =1.93 ± 0.14 µm, ξ(Edom) = 1.38 ± 0.22 µm; diagonal-ization: ξ(Ecent) = 1.18 µm, ξ(Edom) = 0.69 µm): therobust prediction is the hierarchy and the ratio, both re-produced consistently.Figures 8 and 9 summarize the results. Figure 8traces the emergence chain from the disorder landscape[Fig. 8(a)], through the eigenstate PL spectra [Fig. 8(b)],to the δEcent [Fig. 8(c)] and δEdom [Fig. 8(d)] descriptormaps, whose spatial correlation functions and 1/e lengthsare compared in Fig. 9(a) and Fig. 9(b), respectively.c. Two-fluid spectral structure at the single-pixellevel. Figure 8(b) demonstrates that the broad-background plus sharp-peak structure emerges directlyfrom the eigenstate decomposition, without the phe-nomenological amplitude assignment used in Route2. Background-dominated pixels show a broad quasi-Gaussian envelope from ∼20 background eigenstates peroptical spot each broadened by ηbg = 15 meV. Mixed andtrap-rich pixels show narrow peaks (η = 1.5 meV) fromindividual trap eigenstates rising above the broad enve-lope. This two-fluid coexistence is a direct consequenceof the spatial-energy structure of the Hamiltonian eigen-states, not of phenomenological amplitude assignment.220 2 4 6 8x ( m)012345678y (m)Ws = 6 meV, s = 2 m, Nt = 133(a) Disorder landscape Vslow(r)1050510meV60 40 20 0 20E (meV)PL intensity (norm., stacked)  Ecent Edom(b) PL spectra from Hamiltonian eigenstatesbg-dominated (FS=0.00)mixed (FS=0.78)trap-rich (FS=1.00)0 2 4 6 8x ( m)02468y (m)smooth landscape(c) Ecent  ( = 1.18 m)21012E (meV)0 2 4 6 8x ( m)02468trap-switching fluctuations(d) Edom  ( = 0.69 m)21012E (meV)(Ecent) > (Edom)  [1.18 > 0.69 m]    emerges from eigenvalue statisticsFIG. 8. Emergence chain: H → eigenstates → local spectra → descriptor maps. All panels derive from the samediagonalization of H = −(ℏ2/2M)∇2 +Vslow +Vtrap on a 128×128, 8×8 µm grid (best-fit parameters). (a) Disorder landscapeVslow with trap positions (gold ×). Squares, triangles, and circles mark the three representative measurement pixels. Rings showthe 1/e radius of the optical spot (FWHM = 0.85 µm). (b) PL spectra from Hamiltonian eigenstates at the three representativepixels. Filled areas show background-eigenstate contribution (shaded, broad) and trap-eigenstate contribution (solid, narrow);dashed/dotted vertical lines mark Ecent and Edom. The trap-rich pixel shown (FS = 1) is an extreme representative of thediagonalization route, chosen to illustrate the trap-dominated limit; it is not the typical sharp fraction, which averages to theexperimental range ⟨FS⟩ ≈ 0.15–0.40 over the full map. (c) Ecent spatial map (ξ = 1.18 µm): smoothly tracks Vslow. (d)Edom spatial map (ξ = 0.69 µm): more fragmented due to trap-level switching. The annotation confirms ξ(Ecent) > ξ(Edom)emerging from eigenvalue statistics without relying on the synthetic two-fluid spectrum construction.XII. DISCUSSIONA. Limitations of the Minimal ModelThe present model intentionally omits valley physics(K and K′ exciton species) [14, 68–70], spin struc-ture and bright/dark exciton mixing, phonon-assistedemission and momentum-indirect transitions [71, 72],exciton-exciton interactions, and nonequilibrium relax-ation dynamics [73, 74]—all of which are secondary tothe disorder-induced spectral organization at the spatialscales of interest.The justification for this omission is structural ratherthan incidental: valley polarization, bright–dark mix-ing, and phonon-assisted sidebands act within the single-pixel emission envelope, generating fine structure that230 1 2 3 4Distance r ( m)0.20.00.20.40.60.81.0Correlation C(r)1/e(a) Spatial correlations from H diagonalizationEcent ( = 1.18 m)Edom ( = 0.69 m)FS ( = 1.53 m)R1 ( = 1.28 m)Ecent Edom FS R10.00.51.01.52.02.5Correlation length  (m)(Ecent) > (Edom)1.180.691.531.282.001.272.02 2.05(b)  hierarchy: diagonalization vs experimentDiagonalizationExperimentFIG. 9. Correlation-length hierarchy from Hamiltonian diagonalization. (a) Spatial correlation functions C(r) of fourspectral descriptors computed from the 400-pixel map generated by diagonalizing H. Ecent (blue circles) decays more slowlythan Edom (red squares), confirming ξ(Ecent) = 1.18 µm > ξ(Edom) = 0.69 µm. The double arrow marks the hierarchy gap∆ξ. (b) Bar chart comparing diagonalization results (blue) with experimental values (red) of Ref. [35] for the four descriptorswith reported correlation lengths. The hierarchy ξ(Ecent) > ξ(Edom) is reproduced by both approaches, with diagonalizationgiving ratio 0.58 vs. experimental 0.64.can shift, split, or skew individual peaks. They do not,however, modify the spatial low-pass filtering of Vslow bythe optical spot, which is what sets the macroscopic sta-tistical moments—the centroid Ecent and the correlation-length hierarchy of Proposition 3. Their leading ef-fect therefore falls on the fine-structure descriptors (FS ,R1, and the asymmetry encoded in ∆Ecd, RHL), wherethey may renormalize the prefactors of the variance rela-tions (C5), while leaving the spatial organization and itscorrelation-length hierarchy—the central results of thiswork—intact.Several of these approximations can become quanti-tatively important in specific regimes. Valley-coherenteffects may become relevant at very low trap densityand temperatures below ∼ 5 K [14, 69], while highpump-power conditions introduce additional inhomo-geneity through exciton-exciton interactions. The Gaus-sian correlation kernel for Vslow also fails to capture non-Gaussian disorder such as the sharp domain walls arisingfrom moiré reconstruction.A more fundamental limitation is the low trap den-sity nt = 2 µm−2, which places only ∼2–3 traps withineach optical spot and allows Poisson counting noise tosuppress the simulated ξ(FS) and ξ(R1) (≈ 0.6–0.9 µm)well below the experimental values (≈ 2 µm). In the realmoiré system the effective trap density is∼ 103–104 timeshigher, since all moiré sites contribute, and one expectsξ(FS) → ξs in this high-density limit. The clusteringmodel (58) partially mitigates this discrepancy by cor-relating trap positions with Vslow minima, but does notfully close the gap at experimentally relevant densities.B. Non-Gaussian Slow DisorderReal moiré heterobilayers often exhibit reconstructioninto triangular domains with sharp boundaries [52, 53,56]. In such systems, Vslow may be better modeled as apiecewise-constant field with sharp domain walls ratherthan a Gaussian random field.The domain-wall model predicts step-like Ecent mapswith sharp jumps between domain energies, bimodal ortrimodal descriptor distributions in place of the smoothdistributions expected for Gaussian disorder, and spec-tral family boundaries that coincide spatially with thedomain walls.This is a testable distinction: Gaussian disorder givessmooth spectral maps, while reconstruction domains givestep-like maps.A distinct, milder generalization concerns the shapeof the correlation kernel rather than the field statistics.The Gaussian kernel of Eq. (4) is the natural choice forslow disorder arising from long-wavelength strain andelectrostatic fluctuations, but if the landscape is insteaddominated by extended dislocation lines or domain-wallnetworks the correlation is better described by an expo-nential kernel Cs(r) = e−r/ξs . Importantly, the centralsmoothing result—that Ecent is an optically filtered copyof Vslow (Corollary 1) and that the correlation-length hi-erarchy ξ(Ecent) ≥ ξ(Edom) holds (Proposition 3)—is in-sensitive to this choice. Convolution with the optical spotis a low-pass filter that suppresses high spatial frequen-cies regardless of the kernel shape, so the qualitative filterstructure is unchanged: only the precise functional form24of C(r) near the origin, and hence the numerical prefactorrelating ξ(Ecent) to ξs, depends on whether the kernel isGaussian or exponential. The disorder-filter framework istherefore robust to the detailed form of the slow-disordercorrelations.C. Microscopic Origin and Magnitude of theEffective ParametersThe effective disorder amplitudes inferred here—aslow-disorder amplitude Ws ∼ 10–15 meV and an ef-fective trap-switching amplitude W efft ∼ 12 meV—arenot free of microscopic constraints: they should be com-patible with realistic mechanisms in a MoSe2/WSe2 het-erobilayer. Although a full first-principles evaluation isbeyond the scope of this effective theory, the required en-ergy scales are consistent with reported ranges for thesematerials, as summarized in Table IV. Micron-scale vari-ations of the local twist angle and long-wavelength het-erostrain shift the interlayer-exciton energy through thedeformation potential by several to a few tens of meV[29, 54, 56]; moiré reconstruction and stacking-registryvariations produce stacking-dependent exciton shifts of10–50 meV [52, 53, 56]; and dielectric and electrostatic in-homogeneity contributes an additional few to tens of meV[12, 72]; hyperspectral PL imaging has recently begun tomap these strain and disorder fields directly [36]. Super-posed over a micron correlation length these combine to aslow-disorder amplitude of order 10 meV, consistent withthe inferred Ws. The sharp emitters—moiré-trapped ex-citons, atomic defects, and strain-localized sites—carrybinding energies of 5–50 meV [16, 26, 45], so the effec-tive trap-switching amplitude W efft ∼ 12 meV can beaccounted for by a moderate portion of the reportedlocalized-state energy scale. The inferred parametersare therefore consistent with known microscopic energyscales.The bias energy Ebias that co-localizes traps with theslow-potential minima [Eq. (58)] is an energy scale (nota temperature) that sets how strongly the trap densityis biased toward low-energy regions of Vslow. Its mean-ing is fixed by its two limits: as Ebias → ∞ the biasingfactor exp[−Vslow/Ebias] becomes flat and the traps re-duce to an uncorrelated Poisson distribution (no cluster-ing), while for Ebias ≪ Ws the traps are pulled almostentirely into the deepest minima of Vslow (extreme clus-tering). The best-fit Ebias ≈ 7 meV is comparable to Wsand thus lies between these extremes: an energy changeof order Ebias alters the local trap density by a factor∼ e, so the fit describes a moderate bias in which de-fect and strain-trap formation is enhanced—but not ex-clusively confined—at reconstruction domains and strainconcentrators. This correlation energy is of the sameorder as the strain, reconstruction, dielectric, and elec-trostatic scales collected in Table IV, and so is physicallyreasonable for a MoSe2/WSe2 heterobilayer.D. Toward Quantitative Parameter ExtractionA key practical strength of the descriptor-filter frame-work is that it provides a peak-decomposition-free route toinfer effective disorder parameters from hyperspectral PLdata, without microscopic line assignment. The proce-dure maps four observable statistics to four model param-eters sequentially. The slow-disorder correlation lengthξs follows directly from Corollary 1: ξs ≈ ξ(Ecent) =2.00 µm for the experimental system. The slow-disorder amplitude is then given by Ws ≈ Var(Ecent)1/2(Eq. (31)), yielding Ws ∼ 10–15 meV from the observedcentroid energy variation across the map. The relativetrap strength follows from the Ecent–Edom correlation viaEq. (49), using the experimental Spearman value ρS asan estimate of the analytical Pearson coefficient ρP. Thequantity the descriptor covariance determines directly isthe effective trap-switching fluctuation σδ, which is thedominant-energy fluctuation induced by the traps ratherthan the bare trap-depth standard deviation; denotingthis effective trap amplitude W efft ≡ σδ givesσδWs=√1ρP(Ecent, Edom)2− 1 ≡ W efftWs, (63)yielding σδ/Ws ≈ 0.8 from the experimental ρS =0.788; the independent estimate from the correlation-length ratio ξ(Edom)/ξ(Ecent) = 0.64 gives σδ/Ws ≈ 1.1(σ2δ/W2s ≈ 1.2) via the Gaussian relation Eq. (44).Both place σδ at order Ws, consistent with an effec-tive W efft ≳ Ws (experimental values from Ref. [35]).The four-parameter extraction quotes this W efft as theeffective trap amplitude Wt. Finally, the effective sharp-emitter density nt is constrained by the mean sharp frac-tion ⟨FS⟩ ≈ 0.15–0.40, which grows with ntW2t σ2t ; cal-ibrating to ⟨FS⟩ ≈ 0.25 at best-fit Wt = 12 meV andσt = 50 nm gives nt ≈ 2 µm−2. Note that nt is an effec-tive parameter representing the density of optically re-solvable sharp-emitter sites under the experimental con-ditions, not a direct count of microscopic defects (whichwould be orders of magnitude higher in a moiré lat-tice). This effective density also carries an excitation-and temperature-dependence: under higher pump powera larger fraction of trap and moiré sites becomes opti-cally populated, so nefft grows with the exciton filling fac-tor and ultimately saturates as state filling exhausts theavailable sites—bounded from above by the ∼ 103–104moiré sites contained within each optical spot. At lowtemperature, conversely, rapid relaxation funnels exci-tons into the deepest traps and reduces the number of dis-tinct resolvable sharp emitters. The value nt ≈ 2 µm−2should therefore be read as the effective sharp-emitterdensity specific to the excitation power and temperatureof the experiment, rather than as a fixed material con-stant.The four extracted parameters (ξs,Ws,Wt, nt) spec-ify the leading disorder sector of the model and can becross-validated by checking whether the predicted de-25TABLE IV. Characteristic energy scales in MoSe2/WSe2 heterobilayers that contribute to the effective disorder amplitudes.The inferred slow-disorder amplitude Ws ∼ 10–15 meV and effective trap-switching amplitude W efft ∼ 12 meV lie within theseknown ranges.Mechanism Energy scale Contributes toLocal twist-angle variation δθ few–10 meV VslowHeterostrain ε ∼ 0.1–1% few–tens meV VslowReconstruction / stacking registry 10–50 meV Vslow, trapsDielectric / electrostatic disorder few–tens meV VslowLocalized defect / strain traps 5–50 meV Vtrapscriptor correlation length ratio and Spearman matrixreproduce the observed values—an internal consistencycheck that does not require any additional input. Theremaining quantities entering the simulation—the trapradius σt, the broadenings η and ηbg, and the trap-clustering parameter Ebias—are not part of this minimalfour-parameter extraction: σt enters only through thecombination ntW2t σ2t (Sec. VII) and is fixed at a repre-sentative sub-optical value, while Ebias controls the spa-tial co-localization of traps with Vslow and is separatelyconstrained by the correlation lengths of FS and R1. Thisparameter extraction protocol should be applicable toTMD heterobilayer systems imaged by hyperspectral PLmapping [1, 2, 25], providing a spectroscopic disorder di-agnostic independent of microscopic peak identification.E. Generic Versus Moiré-Specific ContentIt is important to delineate which of our results aregeneric to any multi-scale-disordered emitter and whichare specific to a moiré heterobilayer. The descriptor-filtermechanism itself is general: the correlation-length hierar-chy ξ(Ecent) ≥ ξ(Edom), the ∆Ecd–RHL anticorrelation,and the descriptor covariance structure follow from thecoexistence of a smooth, micron-correlated backgroundand a dense set of sharp, sub-spot emitters, and wouldarise equally in a monolayer TMD or a disordered quan-tum well with the same two-scale spectral content. In-deed, the underlying disorder-localized COM-exciton pic-ture and the optically filtered correlation functions areessentially those developed for quantum-well excitons[61–63]; in that sense the framework is a spatially re-solved, descriptor-based counterpart of the mobility-edgephenomenology known there. The energy-dependentlocalization it implies has recently been observed di-rectly in the same material system as a localisation-to-delocalisation transition of moiré excitons [75, 76], andlocalized interlayer excitons occur in MoSe2/WSe2 evenin the absence of a moiré potential [77]—underscoringthat the localization physics underlying our descriptorsis generic rather than moiré-specific.What the moiré heterobilayer provides is a natural andespecially rich realization of the required two-scale struc-ture. First, the moiré superlattice can provide a densemanifold of localized interlayer-exciton states and a highareal density of optically active sites—of order 103–104within a single optical spot—so that the sharp-emitterensemble needed for FS , R1, and the trap-switching fluc-tuation σδ need not rely on sparse extrinsic defects. Sec-ond, moiré reconstruction, stacking-registry variation,and strain can correlate the slow potential with the lo-cations or activation of trap-like emitters, providing aplausible microscopic route to the trap–Vslow clustering[Eq. (58)] that lengthens ξ(FS) and ξ(R1) and can con-tribute to the observed spectral-family structure. Atthe spatial scales analyzed here the moiré potential thusenters principally by setting the density and cluster-ing of the local excitonic manifold (and by renormaliz-ing the effective mass and local DOS), rather than byimprinting its own ∼ 10–20 nm periodicity on the de-scriptors, which is averaged over by the optical spot.More specifically moiré-related signatures—as opposed togeneric multi-scale disorder—would include the triangu-lar reconstruction-domain organization of the descriptormaps and recurrent peaks at the moiré miniband spacing,both accessible only below the optical-spot scale (near-field or low-temperature single-site spectroscopy), as dis-cussed next.F. Extensions: Moiré Potential and QuantumEmitter RegimeWhen Vmoiré is included, the model predicts additionalstructure. The moiré potential introduces a period-aMmodulation of the local DOS, which may manifest asrecurrent spectral peaks at energies separated by themoiré miniband spacing. When Vm exceeds the rele-vant thermal and linewidth scales, individual moiré sitescan host localized quantum-emitter-like exciton states[30, 32, 46, 47, 78], and their spatial brightness distri-bution would then be modulated by the slow disorderfield Vslow—a testable prediction accessible via near-fieldor variable-temperature PL mapping.26TABLE V. Disorder parameter extraction from experimental observables. All quantities are derived from the de-scriptor covariance structure without microscopic line assignment. Here Wt is inferred as an effective trap-switching amplitudeW efft ≡ σδ (the dominant-energy fluctuation induced by the traps), not as a microscopic trap-depth distribution width. The lastrow requires calibration within the two-component (background plus trap) PL model, because ⟨FS⟩ depends on the broadeningsand the relative oscillator strengths in addition to the trap density (Sec. XI D).Observable Extracted parameter Formula Valueξ(Ecent) ξs direct ≈ 2.00 µmVar(Ecent)1/2 Ws Eq. (31) ∼ 10–15 meVρS(Ecent, Edom) σδ/Ws (eff. Wt/Ws) Eq. (63) ≈ 0.8⟨FS⟩ ntW2t σ2t calibration nt ≈ 2 µm−2XIII. CONCLUSIONWe have developed a minimal theory for the spatialorganization of spectral descriptors in moiré exciton pho-toluminescence, using the descriptor-based disorder-filterpicture as an organizing principle for hyperspectral PLdata. Starting from an effective exciton Hamiltonian andthe chain H → ρ(E,R) → I(E,R) → descriptor fields,the paper establishes the following results.The descriptor-based disorder-filter principle pro-vides the unifying framework: each spectral descriptoris preferentially sensitive to a different component of themulti-scale disorder landscape. The centroid energy in-tegrates over the optical spot, filtering out sub-spot trapfluctuations, so that it primarily tracks the slow disorderlandscape. The dominant-peak energy, by contrast, isadditionally randomized by discrete trap-level switchingwithin the optical spot. This differential filtering pro-vides a microscopic mechanism for the observed inter-descriptor differences and the correlation-length hierar-chy.The correlation length hierarchy ξ(Ecent) ≥ξ(Edom) is established analytically from the decompo-sition Edom = V̄s + δEdom. The experimental ratioξ(Edom)/ξ(Ecent) ≈ 0.64 [35] implies an effective trap-switching fluctuation σδ/Ws ≈ 1.1 (of order unity), andHamiltonian diagonalization provides an independentmodel-level check of the hierarchy from the eigenvaluestatistics: ξ(Ecent) = 1.18 µm > ξ(Edom) = 0.69 µm.The sign and approximate magnitude of theprincipal Spearman inter-descriptor correlationsare derived analytically. The near-perfect ∆Ecd–RHLanti-correlation (measured ρS = −0.978 [35]) is a robustspectral shape relation that follows from the geometry ofa broad class of spectra with a dominant unimodal en-velope, while the positive FS–R1 correlation (measuredρS = +0.901 [35]) reflects both descriptors measuring thesame trap-peak density. The calibrated phenomenolog-ical PL simulations reproduce all four principal correla-tions simultaneously, with a mean absolute deviation of0.010 per pair.Spectral families are interpreted as correlated disor-der domains of the Gaussian slow disorder field, ratherthan as thermodynamically distinct phases, with the pre-dicted family count consistent with the experimentallyobserved three families [35].Four qualitatively distinct disorder regimes are iden-tified on a two-dimensional disorder parameter map. Theexperimental system falls in the hierarchically disorderedregime where both slow and trap disorder exceed the ho-mogeneous linewidth, producing multi-scale spectral or-ganization reflected in the descriptor correlation-lengthhierarchy.The framework provides a peak-decomposition-freeroute to constrain the leading effective disorder parame-ters (ξs,Ws,Wt, nt) from hyperspectral PL data via thedescriptor covariance structure, without any microscopicline assignment (Table V). The predicted correlation-length hierarchy and inter-descriptor Spearman matrixconstitute directly testable signatures that can distin-guish disorder regimes and guide future experimentswith higher spatial resolution or variable temperature.Although motivated by MoSe2/WSe2, the frameworkshould be applicable to hyperspectral PL maps in whichspectra contain a smooth envelope together with unre-solved local fine structure; descriptor covariance thenserves as a general spectroscopic disorder diagnostic fortwo-dimensional materials, quantum-dot arrays, and dis-ordered semiconductor heterostructures. Future workshould address valley physics, moiré reconstruction do-main walls, and nonequilibrium relaxation in modifyingthe spectral landscape.ACKNOWLEDGMENTSWe thank R. Kitaura for stimulating discussions onhyperspectral PL organization in moiré heterobilay-ers. This work was supported by JSPS KAKENHI(Grants No. JP25K01609, No. JP22H05473, and No.JP21H01019), JST CREST (Grant No. JPMJCR19T1).K.W. acknowledges the financial support for Basic Sci-ence Research Projects (Grant No. 2401203) from theSumitomo Foundation.27DATA AVAILABILITYThe analysis scripts and numerical data that supportthe findings of this study are available from the corre-sponding author upon reasonable request. The experi-mental data used for comparison are available in Ref. [35].Appendix A: Gaussian Random Field GenerationThe slow potential Vslow is a Gaussian random fieldwith zero mean and a prescribed two-point correlation⟨V (r)V (r′)⟩ = W 2C(|r − r′|). We generate it with thespectral (Fourier-filtering) method, which is exact andcosts only O(N logN). The method rests on the Wiener–Khinchin theorem: for a stationary field the power spec-tral density is the Fourier transform of the autocorrela-tion, S(k) =∫d2rC(r) e−ik·r. Coloring spectrally flat(white) noise with√S(k) therefore imprints preciselythe target correlation C(r). Concretely:1. Generate a white noise field ξ(r) with ⟨ξ(r)ξ(r′)⟩ =δ(2)(r− r′).2. Compute its Fourier transform ξ̃(k).3. Multiply by the square root of the power spectrum,Ṽ (k) =W√S(k) ξ̃(k).4. Inverse Fourier transform: V (r) = IFFT[Ṽ (k)].For the Gaussian kernel C(r) = e−r2/2ξ2s used through-out, the power spectrum is itself Gaussian, S(k) =2πξ2s e−ξ2sk2/2, so the generated field is smooth on thescale ξs.Two implementation points are worth noting. First,the discrete transform carries grid-dependent prefactors(factors of NxNy and the cell area dx2); rather thantracking these, we rescale the output field to its exacttarget standard deviation, V →W V/std(V ), which fixesthe amplitude W unambiguously. Second, the FFT im-poses periodic boundary conditions; because the map sat-isfies L≫ ξs, the resulting wrap-around correlations arenegligible.Appendix B: Finite-Difference HamiltonianWe discretize the exciton Hamiltonian H =−(ℏ2/2M)∇2 + Veff on a 2D lattice with grid spacinga and Nx ×Ny sites. The Laplacian is approximated bythe standard second-order five-point stencil,− ℏ22M∇2ψ(i, j) ≈ ℏ22Ma2[4ψ(i, j)− ψ(i+1, j)− ψ(i−1, j)− ψ(i, j+1)− ψ(i, j−1)]. (B1)Assembling this together with the potential, which is di-agonal in the site basis, gives a sparse Hamiltonian: eachsite (i, j) carries a diagonal entry 4thop + Veff(i, j) and iscoupled to its four nearest neighbors by a hopping −thop,where thop = ℏ2/(2Ma2) is the kinetic (hopping) energyscale. The matrix has dimension NxNy with at most fivenon-zero entries per row. We impose periodic bound-ary conditions, consistent with the periodic slow-disorderfield of Appendix A.For the simulation parameters M = 1.2m0 and ∆x =62.5 nm (a 128 × 128 lattice on an 8 µm domain),thop = ℏ2/(2M∆x2) ≈ 0.0081 meV, consistent withSec. II. Because thop ≪ Ws ∼ 10 meV, the disorder en-ergy dominates the kinetic energy at the grid scale; thekinetic term therefore plays the role of the coarse-grainedspectral generator described in Sec. XI F rather than gov-erning eigenstate localization.Only the lowest optically active eigenstates are re-quired. We obtain them with the Lanczos algorithm(ARPACK, smallest-algebraic mode) [64], as described inSec. XI; for larger grids a shift-and-invert sparse eigen-solver (SLEPc/PETSc) [65, 66] targets the same low-lying band more economically.Appendix C: Spectral Descriptor Variance fromDisorder TheoryFor completeness, we collect the leading-order descrip-tor variances derived in the main text. They split intotwo groups, according to which part of the disorder eachdescriptor measures.The energy-position descriptors follow from the decom-position Edom = V̄s+δEdom (Secs. IVB and VI). The cen-troid tracks the optically filtered slow potential, whereasthe dominant-peak energy carries the additional trap-switching fluctuation σδ; because both share the sameV̄s, it cancels in their difference:Var(Ecent) ≈W 2s , (C1)Var(Edom) ≈W 2s + σ2δ , (C2)Var(∆Ecd) ≈ σ2δ , (C3)Var(RHL) ∝ σ2δ . (C4)The fine-structure descriptors instead measure the traploading within the optical spot. They scale with the com-bination Γt = ntW2t σ2t (Sec. VII), reduced by the spotarea σ2opt (Sec. IV):Var(FS) ∝ ntW2t σ2t /σ2opt, (C5)Var(R1) ∝ ntW2t σ2t /(η2σ2opt), (C6)Var(Sspec) ∝ ntW2t σ2t /σ2opt. (C7)These relations give the scaling of the descriptor vari-ances with the disorder parameters, providing addi-tional testable predictions beyond the spatial correlationlengths. For the dimensionless descriptors (RHL, FS ,28Sspec) and for R1 (of dimension inverse energy), the pro-portionality constants carry the residual dimensions—fixed by the spectral envelope width and the broaden-ing η—so these last four relations express the parameterscaling rather than dimensionally complete identities.Appendix D: Robustness to the Trap Width σtTo verify that the effective trap width σt does not con-trol the main results—as argued in Sec. II, where σt en-ters only through the degenerate combination ntW2t σ2tand through the non-binding hierarchy floor ξt =√2σt—we repeat the full Hamiltonian diagonalization (Sec. XI)while sweeping σt over 50–125 nm, holding the disorderrealization (slow field, trap positions, and trap depths)fixed through common random seeds. Table VI reportsthe outcome in two modes. In the uncompensated sweep(nt fixed), increasing σt deepens and broadens the wells,raising the trap loading (and hence ⟨FS⟩ and the trap-induced variance), yet the correlation-length hierarchyξ(Ecent) > ξ(Edom) and the two principal shape corre-lations ρS(∆Ecd, RHL) ≈ −0.95 and ρS(FS , R1) ≈ +0.93are unchanged. The Ecent–Edom correlation, by contrast,is itself loading-dependent and is not stabilized in thisuncompensated sweep; its anomalously small value atσt = 125 nm (ρEE = −0.01) reflects overloading of thebroadened trap wells and lies outside the best-fit regime,and is included only as a stress test. In the compen-sated sweep, where nt ∝ σ−2t holds the combinationntσ2t fixed, the loading-dependent quantities ⟨FS⟩ andρS(Ecent, Edom) also stabilize, directly confirming the nt–σt degeneracy. In all eight cases the hierarchy orderingis preserved; only the magnitude of the gap narrows asσt grows (the trap-induced fluctuation becomes spatiallybroader, lengthening ξ(Edom) toward ξ(Ecent)), never in-verting.Appendix E: Robustness to the Optical-WeightModelThe local PL weight of a COM eigenstate is, physically,the coherent radiative (oscillator-strength) matrix ele-ment An(R) = |∫Wamp(r−R)ψn(r) d2r|2 of Eq. (13) [61–63], rather than the incoherent local density∫Wopt|ψn|2used in the transparent LDOS form (16). Here we verifythat the descriptor statistics are insensitive to this choice.Using the same Hamiltonian eigenstates (Sec. XID), werecompute all descriptors with both weightings for thesame 128×128, 8×8 µm2 realizations, averaged over fiveindependent disorder seeds. As shown in Fig. 10, everydescriptor statistic—ξ(Ecent), ξ(Edom), their ratio, thethree principal Spearman correlations, and ⟨FS⟩—agreesbetween the two weightings to well within the seed-to-seed scatter (the coherent and incoherent values differ by≤ 0.01 in every case, e.g. ρS(∆Ecd, RHL) = −0.928 vs.−0.929 and ξ(Ecent) = 1.143 vs. 1.142 µm). The reasonσt nt ξcent ξdom ρcd ρFR ρEE(nm) (µm−2) (µm) (µm)Uncompensated (nt = 2.0 fixed)50 2.00 1.18 0.69 −0.97 +0.94 +0.5175 2.00 1.36 1.04 −0.93 +0.96 +0.70100 2.00 1.16 0.87 −0.96 +0.94 +0.45125 2.00 1.21 1.07 −0.98 +0.93 −0.01Compensated (nt ∝ σ−2t )50 2.00 1.18 0.69 −0.97 +0.94 +0.5175 0.89 0.98 0.71 −0.96 +0.92 +0.62100 0.50 1.15 1.03 −0.98 +0.85 +0.82125 0.32 1.18 1.11 −0.98 +0.86 +0.65TABLE VI. Robustness of the diagonalization results to thetrap width σt. ρcd ≡ ρS(∆Ecd, RHL), ρFR ≡ ρS(FS , R1), andρEE ≡ ρS(Ecent, Edom). The hierarchy ξ(Ecent) > ξ(Edom)holds in every row. The compensated sweep keeps ntσ2t fixed,stabilizing the loading-dependent observables ⟨FS⟩ and ρEE ,and thereby confirming the ntW2t σ2t degeneracy.1.0 0.5 0.0 0.5 1.0incoherent weight  Wopt| n|21.00.50.00.51.0coherent weight  |Wampn|2Descriptor statistics: optical-weight choicey = x(Ecent)(Edom)d/ cEcd, RHLFS, R1Ec, EdFSFIG. 10. Descriptor statistics are insensitive to theoptical-weight model. For each descriptor statistic (la-beled), the value obtained with the coherent radiative weight|∫Wampψn|2 (vertical axis) is plotted against that from theincoherent local-density weight∫Wopt|ψn|2 (horizontal axis);error bars are the seed-to-seed standard deviation over fivedisorder realizations. All points lie close to the y = x line,showing that the two weightings yield statistically indistin-guishable descriptor covariance and correlation-length hierar-chy within the seed-to-seed uncertainty.is structural: both weights are localized within the opti-cal spot, so the spatial statistics of the descriptor fields—and hence the correlation-length hierarchy and descriptorcovariance that are the subject of this work—are set bythe disorder landscape, not by the detailed form of theper-state emission weight. The coherent weight does re-distribute intensity toward the nodeless localized (trap)states, but this common re-weighting cancels in the de-scriptor correlations.29Appendix F: Finite-Size, Finite-Step, andFinite-Spot Robustness of the HierarchyThe experimental map spans 8 × 8 µm2, only a fewtimes the correlation lengths, so the extracted 1/e lengthscarry finite-size uncertainty. We test the robustnessof the hierarchy ξ(Ecent) > ξ(Edom) by sweeping themap size L, the scan pitch, and the optical-spot width,each over thirty disorder realizations, using the fast phe-nomenological descriptor generator (Sec. XID) so thatmany configurations can be sampled. Figure 11 sum-marizes the domain-size sweep, showing the correlationlengths ξ(Ecent) and ξ(Edom) versus map size [Fig. 11(a)]and the finite-size-robust hierarchy ratio [Fig. 11(b)].Two features stand out. First, in this phenomenologi-cal sweep the hierarchy holds across the sampled con-figurations, and the ratio ξ(Edom)/ξ(Ecent) is much lesssensitive to sampling than the absolute lengths: it staysat 0.69–0.70 across L = 8–20 µm and scan pitches 0.25–0.60 µm, and varies only mildly (from 0.74 to 0.64) as theoptical FWHM is swept over 0.60–1.10 µm, all bracket-ing the experimental ratio 0.64. Second, the absolute 1/elengths are biased low on the smallest map and drift up-ward with L (a finite-size estimator bias); at the experi-mental map size the simulation reproduces the measuredvalues (ξ(Ecent) = 1.92 ± 0.17 µm vs. 2.00; ξ(Edom) =1.33 ± 0.25 µm vs. 1.27). These differ slightly from theFig. 3 ensemble (1.93 and 1.38 µm) because the finite-size sweep uses an independently generated set of realiza-tions. Because finite sampling biases both lengths in thesame direction, the physically meaningful ratio is com-paratively stable. The principal anticorrelation is like-wise robust, ρS(∆Ecd, RHL) = −0.97± 0.01 throughout.This supports the conclusion that the correlation-lengthhierarchy is not primarily a finite-sampling or coarse-gridding artifact, consistent with the ensemble-averageinequality of Proposition 3; occasional per-realization in-versions seen in the microscopic diagonalization on a sin-gle 8 µm map are the expected finite-size scatter aboutthis ensemble result.[1] E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H.MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani,and A. F. Young, The marvels of moiré materials, Nat.Rev. Mater. 6, 201 (2021).[2] K. F. Mak and J. Shan, Semiconductor moiré materials,Nat. Nanotechnol. 17, 686 (2022).[3] E. C. Regan, D. Wang, E. Y. Paik, Y. Zeng, L. Zhang,J. Zhu, A. H. MacDonald, H. Deng, and F. Wang, Emerg-ing exciton physics in transition metal dichalcogenideheterobilayers, Nat. Rev. Mater. 7, 778 (2022).[4] A. K. Geim and I. V. Grigorieva, Van der Waals het-erostructures, Nature 499, 419 (2013).[5] K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H.Castro Neto, 2D materials and van der Waals het-erostructures, Science 353, aac9439 (2016).[6] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,E. Kaxiras, and P. Jarillo-Herrero, Unconventional super-conductivity in magic-angle graphene superlattices, Na-ture 556, 43 (2018).[7] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-fillingin magic-angle graphene superlattices, Nature 556, 80(2018).[8] D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J.Millis, J. Hone, C. R. Dean, D. N. Basov, A. N. Pa-supathy, and A. Rubio, Moiré heterostructures as acondensed-matter quantum simulator, Nat. Phys. 17, 155(2021).[9] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz,Atomically thin MoS2: a new direct-gap semiconductor,Phys. Rev. Lett. 105, 136805 (2010).[10] A. Splendiani, L. Sun, Y. Zhang, T. Li, J. Kim, C.-Y.Chim, G. Galli, and F. Wang, Emerging photolumines-cence in monolayer MoS2, Nano Lett. 10, 1271 (2010).[11] G. Eda, H. Yamaguchi, D. Voiry, T. Fujita, M. Chen,and M. Chhowalla, Photoluminescence from ChemicallyExfoliated MoS2, Nano Lett. 11, 5111 (2011).[12] G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz,X. Marie, T. Amand, and B. Urbaszek, Colloquium: exci-tons in atomically thin transition metal dichalcogenides,Rev. Mod. Phys. 90, 021001 (2018).[13] K. F. Mak and J. Shan, Photonics and optoelectronics of2D semiconductor transition metal dichalcogenides, Nat.Photonics 10, 216 (2016).[14] J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross,K. L. Seyler, W. Yao, and X. Xu, Valleytronics in 2Dmaterials, Nat. Rev. Mater. 1, 16055 (2016).[15] H. Yu, G.-B. Liu, and W. Yao, Brightened spin-tripletinterlayer excitons and optical selection rules in van derWaals heterobilayers, 2D Mater. 5, 035021 (2018).[16] K. Tran, G. Moody, F. Wu, X. Lu, J. Choi, K. Kim,A. Rai, D. A. Sanchez, J. Quan, A. Singh, J. Emb-ley, A. Zepeda, M. Campbell, T. Autry, T. Taniguchi,K. Watanabe, N. Lu, S. K. Banerjee, K. L. Silverman,S. Kim, E. Tutuc, L. Yang, A. H. MacDonald, and X. Li,Evidence for moiré excitons in van der Waals heterostruc-tures, Nature 567, 71 (2019).[17] C. Jin, E. C. Regan, A. Yan, M. I. B. Utama, D. Wang,S. Zhao, Y. Qin, S. Yang, Z. Zheng, S. Shi, K. Watanabe,T. Taniguchi, S. Tongay, A. Zettl, and F. Wang, Obser-vation of moiré excitons in WSe2/WS2 heterostructuresuperlattices, Nature 567, 76 (2019).[18] Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi,J. Hone, V. Elser, K. F. Mak, and J. Shan, Correlated in-sulating states at fractional fillings of moiré superlattices,Nature 587, 214 (2020).[19] Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak,K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan,and K. F. Mak, Simulation of Hubbard model physics inWSe2/WS2 moiré superlattices, Nature 579, 353 (2020).https://doi.org/10.1038/s41578-021-00284-1https://doi.org/10.1038/s41578-021-00284-1https://doi.org/10.1038/s41565-022-01165-6https://doi.org/10.1038/s41578-022-00440-1https://doi.org/10.1038/nature12385https://doi.org/10.1126/science.aac9439https://doi.org/10.1038/nature26160https://doi.org/10.1038/nature26160https://doi.org/10.1038/nature26154https://doi.org/10.1038/nature26154https://doi.org/10.1038/s41567-020-01154-3https://doi.org/10.1038/s41567-020-01154-3https://doi.org/10.1103/PhysRevLett.105.136805https://doi.org/10.1021/nl903868whttps://doi.org/10.1021/nl201874whttps://doi.org/10.1103/RevModPhys.90.021001https://doi.org/10.1038/nphoton.2015.282https://doi.org/10.1038/nphoton.2015.282https://doi.org/10.1038/natrevmats.2016.55https://doi.org/10.1088/2053-1583/aac065https://doi.org/10.1038/s41586-019-0975-zhttps://doi.org/10.1038/s41586-019-0976-yhttps://doi.org/10.1038/s41586-020-2868-6https://doi.org/10.1038/s41586-020-2085-3300.00.51.01.52.02.53.01/e correlation length (m)(a)  vs. map sizeexp. (Ecent)exp. (Edom)(Ecent)(Edom)8 12 16 20map size L ( m)0.40.50.60.70.80.91.0(Edom)/(Ecent)(b) hierarchy ratio (finite-size robust)exp. ratio = 0.64sim. (Edom)/ (Ecent)FIG. 11. Finite-size behavior of the correlation-lengthhierarchy. (a) 1/e correlation lengths ξ(Ecent) and ξ(Edom)versus map size L (mean ± std over thirty disorder realiza-tions; fixed 0.4 µm pitch and 0.85 µm optical FWHM); filledmarkers are the experimental values at L = 8 µm. (b) Thehierarchy ratio ξ(Edom)/ξ(Ecent) is finite-size-robust (≈ 0.69–0.70) and close to the experimental value 0.64 (dotted line),even though the absolute lengths in (a) drift with L.[20] Y. Shimazaki, I. Schwartz, K. Watanabe, T. Taniguchi,M. Kroner, and A. Imamoglu, Strongly correlated elec-trons and hybrid excitons in a moiré heterostructure, Na-ture 580, 472 (2020).[21] P. Rivera, J. R. Schaibley, A. M. Jones, J. S. Ross, S. Wu,G. Aivazian, P. Klement, K. Seyler, G. Clark, N. J.Ghimire, J. Yan, D. G. Mandrus, W. Yao, and X. Xu,Observation of long-lived interlayer excitons in mono-layer MoSe2–WSe2 heterostructures, Nat. Commun. 6,6242 (2015).[22] P. Rivera, K. L. Seyler, H. Yu, J. R. Schaibley, J. Yan,D. G. Mandrus, W. Yao, and X. Xu, Valley-polarizedexciton dynamics in a 2D semiconductor heterostructure,Science 351, 688 (2016).[23] P. Nagler, G. Plechinger, M. V. Ballottin, A. Mitioglu,S. Meier, N. Paradiso, C. Strunk, A. Chernikov, P. C. M.Christianen, C. Schüller, and T. Korn, Interlayer excitondynamics in a dichalcogenide monolayer heterostructure,2D Mater. 4, 025112 (2017).[24] M. Okada, A. Kutana, Y. Kureishi, Y. Kobayashi,Y. Saito, T. Saito, K. Watanabe, T. Taniguchi, S. Gupta,Y. Miyata, B. I. Yakobson, H. Shinohara, and R. Ki-taura, Direct and Indirect Interlayer Excitons in a van derWaals Heterostructure of hBN/WS2/MoS2/hBN, ACSNano 12, 2498 (2018).[25] A. Ciarrocchi, F. Tagarelli, A. Avsar, and A. Kis, Exci-tonic devices with van der Waals heterostructures: val-leytronics meets twistronics, Nat. Rev. Mater. 7, 449(2022).[26] K. L. Seyler, P. Rivera, H. Yu, N. P. Wilson, E. L. Ray,D. G. Mandrus, J. Yan, W. Yao, and X. Xu, Signaturesof moiré-trapped valley excitons in MoSe2/WSe2 hetero-bilayers, Nature 567, 66 (2019).[27] E. M. Alexeev, D. A. Ruiz-Tijerina, M. Danovich, M. J.Hamer, D. J. Terry, P. K. Nayak, S. Ahn, S. Pak, J. Lee,J. I. Sohn, M. R. Molas, M. Koperski, K. Watanabe,T. Taniguchi, K. S. Novoselov, R. V. Gorbachev, H. S.Shin, V. I. Fal’ko, and A. I. Tartakovskii, Resonantly hy-bridized excitons in moiré superlattices in van der Waalsheterostructures, Nature 567, 81 (2019).[28] S. Brem, C. Linderälv, P. Erhart, and E. Malic, Tunablephases of moiré excitons in van der Waals heterostruc-tures, Nano Lett. 20, 8534 (2020).[29] J. Choi, M. Florian, A. Steinhoff, D. Erben, K. Tran,D. S. Kim, L. Sun, J. Quan, R. Claassen, S. Ma-jumder, J. A. Hollingsworth, T. Taniguchi, K. Watan-abe, K. Ueno, A. Singh, G. Moody, F. Jahnke, and X. Li,Twist angle-dependent interlayer exciton lifetimes in vander Waals heterostructures, Phys. Rev. Lett. 126, 047401(2021).[30] Y.-M. He, G. Clark, J. R. Schaibley, Y. He, M.-C. Chen,Y.-J. Wei, X. Ding, Q. Zhang, W. Yao, X. Xu, C.-Y. Lu,and J.-W. Pan, Single quantum emitters in monolayersemiconductors, Nat. Nanotechnol. 10, 497 (2015).[31] M. Koperski, K. Nogajewski, A. Arora, V. Cherkez,P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, andM. Potemski, Single photon emitters in exfoliated WSe2structures, Nat. Nanotechnol. 10, 503 (2015).[32] A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke,A. Kis, and A. Imamoglu, Optically active quantum dotsin monolayer WSe2, Nat. Nanotechnol. 10, 491 (2015).[33] C. Chakraborty, L. Kinnischtzke, K. M. Goodfellow,R. Beams, and A. N. Vamivakas, Voltage-controlledquantum light from an atomically thin semiconductor,Nat. Nanotechnol. 10, 507 (2015).[34] P. Tonndorf, R. Schmidt, R. Schneider, J. Kern,M. Buscema, G. A. Steele, A. Castellanos-Gomez, H. S. J.van der Zant, S. Michaelis de Vasconcellos, and R. Brats-chitsch, Single-photon emission from localized excitons inan atomically thin semiconductor, Optica 2, 347 (2015).[35] N. F. Ahmad, Y. Urano, K. Watanabe, T. Taniguchi,D. Kozawa, and R. Kitaura, Hierarchical spectralinhomogeneity in photoluminescence of a twistedMoSe2/WSe2 heterobilayer moiré superlattice revealedby hyperspectral mapping, Appl. Phys. Lett. 128, 231901(2026).[36] A. Alfrey, C. Tait, T. Taniguchi, K. Watanabe, and S. T.Cundiff, Revealing strain and disorder in transition-metaldichalcogenides using hyperspectral photoluminescenceimaging (2026), arXiv:2604.01412.[37] F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hub-bard model physics in transition metal dichalcogenidemoiré bands, Phys. Rev. Lett. 121, 026402 (2018).[38] F. Wu, T. Lovorn, and A. H. MacDonald, Topologicalexciton bands in moiré heterojunctions, Phys. Rev. Lett.https://doi.org/10.1038/s41586-020-2191-2https://doi.org/10.1038/s41586-020-2191-2https://doi.org/10.1038/ncomms7242https://doi.org/10.1038/ncomms7242https://doi.org/10.1126/science.aac7820https://doi.org/10.1088/2053-1583/aa7352https://doi.org/10.1021/acsnano.7b08253https://doi.org/10.1021/acsnano.7b08253https://doi.org/10.1038/s41578-021-00408-7https://doi.org/10.1038/s41578-021-00408-7https://doi.org/10.1038/s41586-019-0957-1https://doi.org/10.1038/s41586-019-0986-9https://doi.org/10.1021/acs.nanolett.0c03019https://doi.org/10.1103/PhysRevLett.126.047401https://doi.org/10.1103/PhysRevLett.126.047401https://doi.org/10.1038/nnano.2015.75https://doi.org/10.1038/nnano.2015.67https://doi.org/10.1038/nnano.2015.60https://doi.org/10.1038/nnano.2015.79https://doi.org/10.1364/OPTICA.2.000347https://doi.org/10.1063/5.0335306https://doi.org/10.1063/5.0335306https://arxiv.org/abs/2604.01412https://doi.org/10.1103/PhysRevLett.121.026402https://doi.org/10.1103/PhysRevLett.118.14740131118, 147401 (2017).[39] M. H. Naik, E. C. Regan, Z. Zhang, Y.-H. Chan, Z. Li,D. Wang, Y. Yoon, C. S. Ong, W. Zhao, S. Zhao,M. I. B. Utama, B. Gao, X. Wei, M. Sayyad, K. Yu-migeta, K. Watanabe, T. Taniguchi, S. Tongay, F. H.da Jornada, F. Wang, and S. G. Louie, Intralayer charge-transfer moiré excitons in van der Waals superlattices,Nature 609, 52 (2022).[40] D. A. Ruiz-Tijerina and V. I. Fal’ko, Interlayer hybridiza-tion and moiré superlattice minibands for electrons andexcitons in heterobilayers of transition-metal dichalco-genides, Phys. Rev. B 99, 125424 (2019).[41] M. H. Naik and M. Jain, Ultraflatbands and shear soli-tons in moiré patterns of twisted bilayer transition metaldichalcogenides, Phys. Rev. Lett. 121, 266401 (2018).[42] E. C. Regan, D. Wang, C. Jin, M. I. B. Utama, B. Gao,X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta,M. Blei, J. D. Carlström, K. Watanabe, T. Taniguchi,S. Tongay, M. Crommie, A. Zettl, and F. Wang, Mott andgeneralized Wigner crystal states in WSe2/WS2 moirésuperlattices, Nature 579, 359 (2020).[43] Y. Bai, L. Zhou, J. Wang, W. Wu, L. J. McGilly, D. Hal-bertal, C. F. B. Lo, F. Liu, J. Ardelean, P. Rivera,N. R. Finney, X.-C. Yang, D. N. Basov, W. Yao, X. Xu,J. Hone, A. N. Pasupathy, and X.-Y. Zhu, Excitons instrain-induced one-dimensional moiré potentials at tran-sition metal dichalcogenide heterojunctions, Nat. Mater.19, 1068 (2020).[44] A. Ghiotto, E.-M. Shih, G. S. S. G. Pereira, D. A.Rhodes, B. Kim, J. Zang, A. J. Millis, K. Watanabe,T. Taniguchi, J. C. Hone, L. Wang, C. R. Dean, andA. N. Pasupathy, Quantum criticality in twisted transi-tion metal dichalcogenides, Nature 597, 345 (2021).[45] K. Parto, S. I. Azzam, K. Banerjee, and G. Moody, De-fect and strain engineering of monolayer WSe2 enablessite-controlled single-photon emission up to 150 K, Nat.Commun. 12, 3585 (2021).[46] H. Yu, M. Chen, and W. Yao, Giant magnetic field frommoiré-induced Berry phase in homobilayer semiconduc-tors, Natl. Sci. Rev. 7, 12 (2020).[47] A. Branny, S. Kumar, R. Proux, and B. D. Gerardot,Deterministic strain-induced arrays of quantum emittersin a two-dimensional semiconductor, Nat. Commun. 8,15053 (2017).[48] A. Tarantola, Inverse Problem Theory and Methods forModel Parameter Estimation (Society for Industrial andApplied Mathematics, Philadelphia, 2005).[49] T. C. Berkelbach, M. S. Hybertsen, and D. R. Reich-man, Theory of neutral and charged excitons in mono-layer transition metal dichalcogenides, Phys. Rev. B 88,045318 (2013).[50] A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi,Y. Li, O. B. Aslan, D. R. Reichman, M. S. Hybertsen, andT. F. Heinz, Exciton binding energy and nonhydrogenicRydberg series in monolayer WS2, Phys. Rev. Lett. 113,076802 (2014).[51] H. Yu, G.-B. Liu, J. Tang, X. Xu, and W. Yao, Moiréexcitons: from programmable quantum emitter arraysto spin-orbit–coupled artificial lattices, Sci. Adv. 3,e1701696 (2017).[52] A. Weston, Y. Zou, V. Enaldiev, A. Summerfield,N. Clark, V. Zólyomi, A. Graham, C. Yelgel, S. Magor-rian, M. Zhou, J. Zultak, D. Hopkinson, A. Barinov,T. H. Bointon, A. Kretinin, N. R. Wilson, P. H. Be-ton, V. I. Fal’ko, S. J. Haigh, and R. Gorbachev, Atomicreconstruction in twisted bilayers of transition metaldichalcogenides, Nat. Nanotechnol. 15, 592 (2020).[53] L. J. McGilly, A. Kerelsky, N. R. Finney, K. Shapovalov,E.-M. Shih, A. Ghiotto, Y. Zeng, S. L. Moore, W. Wu,Y. Bai, K. Watanabe, T. Taniguchi, M. Stengel, L. Zhou,J. Hone, X. Zhu, D. N. Basov, C. Dean, C. E. Dreyer,and A. N. Pasupathy, Visualization of moiré superlat-tices, Nat. Nanotechnol. 15, 580 (2020).[54] H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund,S. T. Pantelides, and K. I. Bolotin, Bandgap engineer-ing of strained monolayer and bilayer MoS2, Nano Lett.13, 3626 (2013).[55] A. Raja, L. Waldecker, J. Zipfel, Y. Cho, S. Brem, J. D.Ziegler, M. Kulig, T. Taniguchi, K. Watanabe, E. Malic,T. F. Heinz, T. C. Berkelbach, and A. Chernikov, Di-electric disorder in two-dimensional materials, Nat. Nan-otechnol. 14, 832 (2019).[56] S. Shabani, D. Halbertal, W. Wu, M. Chen, S. Liu,J. Hone, W. Yao, D. N. Basov, X. Zhu, and A. N. Pasu-pathy, Deep moiré potentials in twisted transition metaldichalcogenide bilayers, Nat. Phys. 17, 720 (2021).[57] H.-P. Komsa and A. V. Krasheninnikov, Native defectsin bulk and monolayer MoS2 from first principles, Phys.Rev. B 91, 125304 (2015).[58] N. R. Campbell, The study of discontinuous phenomena,Proc. Cambridge Philos. Soc. 15, 117 (1909).[59] N. R. Campbell, Discontinuities in light emission, Proc.Cambridge Philos. Soc. 15, 310 (1910).[60] H. Haug and S. W. Koch, Quantum Theory of the Opti-cal and Electronic Properties of Semiconductors, 5th ed.(World Scientific, Singapore, 2009).[61] E. Runge and R. Zimmermann, Spatially resolved spec-tra, effective mobility edge, and level repulsion in narrowquantum wells, Phys. Status Solidi B 206, 167 (1998).[62] E. Runge and R. Zimmermann, Level repulsion in exci-tonic spectra of disordered systems and local relaxationkinetics, Ann. Phys. (Leipzig) 510, 417 (1998).[63] R. Zimmermann, E. Runge, and V. Savona, Theory ofresonant secondary emission: Rayleigh scattering versusluminescence, in Quantum Coherence, Correlation andDecoherence in Semiconductor Nanostructures, editedby T. Takagahara (Academic Press, San Diego, 2003)Chap. 4, pp. 89–165.[64] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACKUsers’ Guide: Solution of Large-Scale Eigenvalue Prob-lems with Implicitly Restarted Arnoldi Methods (Societyfor Industrial and Applied Mathematics, Philadelphia,1998).[65] V. Hernandez, J. E. Roman, and V. Vidal, SLEPc: Ascalable and flexible toolkit for the solution of eigenvalueproblems, ACM Trans. Math. Softw. 31, 351 (2005).[66] S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith,Efficient Management of Parallelism in Object-OrientedNumerical Software Libraries, in Modern Software Toolsfor Scientific Computing (Birkhäuser, Boston, 1997) pp.163–202.[67] N. A. C. Cressie, Statistics for Spatial Data (Wiley, NewYork, 1993).[68] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Cou-pled spin and valley physics in monolayers of MoS2 andother group-VI dichalcogenides, Phys. Rev. Lett. 108,196802 (2012).https://doi.org/10.1103/PhysRevLett.118.147401https://doi.org/10.1038/s41586-022-04991-9https://doi.org/10.1103/PhysRevB.99.125424https://doi.org/10.1103/PhysRevLett.121.266401https://doi.org/10.1038/s41586-020-2092-4https://doi.org/10.1038/s41563-020-0730-8https://doi.org/10.1038/s41563-020-0730-8https://doi.org/10.1038/s41586-021-03815-6https://doi.org/10.1038/s41467-021-23709-5https://doi.org/10.1038/s41467-021-23709-5https://doi.org/10.1093/nsr/nwz117https://doi.org/10.1038/ncomms15053https://doi.org/10.1038/ncomms15053https://doi.org/10.1103/PhysRevB.88.045318https://doi.org/10.1103/PhysRevB.88.045318https://doi.org/10.1103/PhysRevLett.113.076802https://doi.org/10.1103/PhysRevLett.113.076802https://doi.org/10.1126/sciadv.1701696https://doi.org/10.1126/sciadv.1701696https://doi.org/10.1038/s41565-020-0682-9https://doi.org/10.1038/s41565-020-0708-3https://doi.org/10.1021/nl4014748https://doi.org/10.1021/nl4014748https://doi.org/10.1038/s41565-019-0520-0https://doi.org/10.1038/s41565-019-0520-0https://doi.org/10.1038/s41567-021-01174-7https://doi.org/10.1103/PhysRevB.91.125304https://doi.org/10.1103/PhysRevB.91.125304https://doi.org/10.1002/(SICI)1521-3951(199803)206:1<167::AID-PSSB167>3.0.CO;2-Lhttps://doi.org/10.1002/andp.199851005-609https://doi.org/10.1137/1.9780898719628https://doi.org/10.1137/1.9780898719628https://doi.org/10.1137/1.9780898719628https://doi.org/10.1145/1089014.1089019https://doi.org/10.1007/978-1-4612-1986-6_8https://doi.org/10.1007/978-1-4612-1986-6_8https://doi.org/10.1103/PhysRevLett.108.196802https://doi.org/10.1103/PhysRevLett.108.19680232[69] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control ofvalley polarization in monolayer MoS2 by optical helicity,Nat. Nanotechnol. 7, 494 (2012).[70] K. Shinokita, K. Watanabe, T. Taniguchi, and K. Mat-suda, Valley Relaxation of the Moiré Excitons in aWSe2/MoSe2 Heterobilayer, ACS Nano 16, 16862 (2022).[71] G. Moody, C. K. Dass, K. Hao, C.-H. Chen, L.-J. Li,A. Singh, K. Tran, G. Clark, X. Xu, G. Berghäuser,E. Malic, A. Knorr, and X. Li, Intrinsic homogeneouslinewidth and broadening mechanisms of excitons inmonolayer transition metal dichalcogenides, Nat. Com-mun. 6, 8315 (2015).[72] F. Cadiz, E. Courtade, C. Robert, G. Wang, Y. Shen,H. Cai, T. Taniguchi, K. Watanabe, H. Carrere, D. La-garde, M. Manca, T. Amand, P. Renucci, S. Tongay,X. Marie, and B. Urbaszek, Excitonic linewidth ap-proaching the homogeneous limit in MoS2-based van derWaals heterostructures, Phys. Rev. X 7, 021026 (2017).[73] M. Kulig, J. Zipfel, P. Nagler, S. Blanter, C. Schüller,T. Korn, N. Paradiso, M. M. Glazov, and A. Chernikov,Exciton diffusion and halo effects in monolayer semicon-ductors, Phys. Rev. Lett. 120, 207401 (2018).[74] J. Zipfel, M. Kulig, R. Perea-Cauśın, S. Brem, J. D.Ziegler, R. Rosati, T. Taniguchi, K. Watanabe, M. M.Glazov, E. Malic, and A. Chernikov, Exciton diffusionin monolayer semiconductors with suppressed disorder,Phys. Rev. B 101, 115430 (2020).[75] E. Blundo, F. Tuzi, S. Cianci, M. Cuccu, K. Olkowska-Pucko,  L. Kipczak, G. Contestabile, A. Miriametro,M. Felici, G. Pettinari, T. Taniguchi, K. Watan-abe, A. Babiński, M. R. Molas, and A. Polimeni,Localisation-to-delocalisation transition of moiré exci-tons in WSe2/MoSe2 heterostructures, Nat. Commun.15, 1057 (2024).[76] H. Fang, Q. Lin, Y. Zhang, J. Thompson, S. Xiao, Z. Sun,E. Malic, S. P. Dash, and W. Wieczorek, Localization andinteraction of interlayer excitons in MoSe2/WSe2 heter-obilayers, Nat. Commun. 14, 6910 (2023).[77] F. Mahdikhanysarvejahany, D. N. Shanks, M. Klein,Q. Wang, M. R. Koehler, D. G. Mandrus, T. Taniguchi,K. Watanabe, O. L. A. Monti, B. J. LeRoy, and J. R.Schaibley, Localized interlayer excitons in MoSe2–WSe2heterostructures without a moiré potential, Nat. Com-mun. 13, 5354 (2022).[78] M. Brotons-Gisbert, B. D. Gerardot, A. W. Holleitner,and U. Wurstbauer, Interlayer and moiré excitons inatomically thin double layers: from individual quantumemitters to degenerate ensembles, MRS Bull. 49, 914(2024).https://doi.org/10.1038/nnano.2012.96https://doi.org/10.1021/acsnano.2c06813https://doi.org/10.1038/ncomms9315https://doi.org/10.1038/ncomms9315https://doi.org/10.1103/PhysRevX.7.021026https://doi.org/10.1103/PhysRevLett.120.207401https://doi.org/10.1103/PhysRevB.101.115430https://doi.org/10.1038/s41467-024-44739-9https://doi.org/10.1038/s41467-024-44739-9https://doi.org/10.1038/s41467-023-42710-8https://doi.org/10.1038/s41467-022-33082-6https://doi.org/10.1038/s41467-022-33082-6https://doi.org/10.1557/s43577-024-00772-zhttps://doi.org/10.1557/s43577-024-00772-z Hierarchical Disorder in Moiré Exciton Photoluminescence Probed by Spectral-Descriptor Correlations Abstract Introduction Model: Effective Exciton Hamiltonian Center-of-Mass Hamiltonian Moiré Potential Long-Wavelength Correlated Disorder Local Trap Disorder Parameter Hierarchy and Scale Separation Green's Function Formalism and Local PL Spectrum Retarded Green's Function Local Density of States Local PL Spectrum Spectral Decomposition: Background and Trap Contributions Optical Coarse-Graining of Slow Disorder Statistical Theory of Spectral Descriptors Definitions of Spectral Descriptors Centroid Energy Dominant-Peak Energy Centroid-Dominant Offset Quantile Width High/Low Ratio Sharp Fraction Spectral Roughness Spectral Entropy Centroid Energy: Slow Disorder Dominance Spectral Entropy and Roughness: Trap Disorder Dominance Correlation Length Hierarchy Main Result Proof Inter-Descriptor Correlation Theory Centroid-Dominant Correlation The Ecd–RHL Anti-Correlation Sharp Fraction – Roughness Correlation Width – Entropy Correlation Disorder Parameter Regimes Control Parameters Regime Boundaries Location of the Experimental System Two-Fluid LDOS Structure Spectral Families as Correlated Disorder Domains Origin of Spectral Families Statistical Nature of Spectral Families Multi-Scale Spectral Organization in the Hierarchical Regime Numerical Simulations Overview: Two Complementary Approaches Trap Clustering Model Parameter Optimization Key Numerical Results Necessity of Hierarchical Disorder: Single-Scale Comparison Continuum LDOS Validation of the Descriptor Hierarchy Discussion Limitations of the Minimal Model Non-Gaussian Slow Disorder Microscopic Origin and Magnitude of the Effective Parameters Toward Quantitative Parameter Extraction Generic Versus Moiré-Specific Content Extensions: Moiré Potential and Quantum Emitter Regime Conclusion Acknowledgments Data Availability Gaussian Random Field Generation Finite-Difference Hamiltonian Spectral Descriptor Variance from Disorder Theory Robustness to the Trap Width t Robustness to the Optical-Weight Model Finite-Size, Finite-Step, and Finite-Spot Robustness of the Hierarchy References