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arxiv: 2605.21254 · v1 · pith:YQYEKNDMnew · submitted 2026-05-20 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech· math-ph· math.MP· physics.data-an

Random Matrix Spectra from Boltzmann-Weighted Lattice Ensembles

Yaprak \"Onder , Abbas Ali Saberi , Roderich Moessner This is my paper

Pith reviewed 2026-05-21 03:39 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mechmath-phmath.MPphysics.data-an
keywords random matrix spectraBoltzmann ensembleslattice correlationsIsing modelspin glassesresolvent formalismWick contractionscritical spectra
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The pith

Boltzmann-sampled lattice configurations map to random matrices whose eigenvalue spectra transition from semicircle law to forms set by spatial correlations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs ensembles of random matrices directly from equilibrium snapshots of lattice models drawn according to the Boltzmann weight. Spatial correlation functions of the lattice system determine the covariance structure of the matrices, which in turn fixes a momentum-space variance profile for the entries. This profile is derived explicitly for both finite correlation length and critical regimes, after which spectral moments are obtained through Wick contractions and the bulk density through a self-consistent resolvent equation. Numerical checks on the two-dimensional Ising model and the three-dimensional Edwards-Anderson spin glass confirm that the spectra depart from the semicircle shape precisely when the underlying correlations become long-ranged.

Core claim

Equilibrium configurations sampled from a Boltzmann measure are mapped to matrix ensembles whose covariance structure is inherited from the spatial correlations of the underlying model, yielding a direct bridge from real-space correlation functions to a momentum-space variance profile; the resulting spectra evolve from the semicircle law at high temperature to model-dependent critical forms that reflect the structure of those correlations.

What carries the argument

The momentum-space variance profile obtained by Fourier-transforming the real-space correlation function of the lattice model, which supplies the second-moment structure for the correlated random-matrix ensemble.

If this is right

  • Spectral moments of the ensemble can be computed order by order using Wick contractions once the variance profile is known.
  • The bulk eigenvalue density satisfies a self-consistent equation obtained from the resolvent formalism.
  • In the high-temperature phase the spectra collapse to the semicircle law for any lattice model.
  • At criticality the spectra acquire shapes that encode the specific decay or power-law form of the spatial correlations.
  • The same construction applies uniformly to both ordered systems such as the Ising model and disordered systems such as the Edwards-Anderson spin glass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework supplies a spectral diagnostic that could detect the onset of long-range correlations without direct measurement of real-space correlation functions.
  • Different universality classes of lattice models should produce distinguishable critical spectra once the variance profile is inserted into the resolvent equation.
  • Finite-size scaling of the matrix spectra near criticality could be compared with known scaling of the correlation length to test consistency.
  • The construction defines a new family of physically motivated correlated random matrices whose statistics are fixed by statistical mechanics rather than by ad-hoc assumptions.

Load-bearing premise

The covariance matrix of each matrix ensemble is taken to be exactly the Fourier transform of the lattice model's spatial correlation function, with no extra model-specific corrections.

What would settle it

Monte Carlo sampling of matrix spectra from the two-dimensional Ising model at criticality that fails to reproduce the analytically predicted departure from the semicircle law.

Figures

Figures reproduced from arXiv: 2605.21254 by Abbas Ali Saberi, Roderich Moessner, Yaprak \"Onder.

Figure 1
Figure 1. Figure 1: FIG. 1: Behavior of [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Distinct pairings (up to a symmetry factor) for the 4th and 6th moments of Wigner matrices. We have [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: ⟨λ 6 ⟩ = 1 Id(ξ) 3 Z Λ4 d 4k (2π) 4 3Gˆ d(k1, k2) Gˆ d(k2, k3) Gˆ d(k3, k4) +2Gˆ d(k1, k2) Gˆ d(k1, k3) Gˆ d(k1, k4), (IV.7) The leading small ξ behaviour of the relevant integrals can be calculated explicitly: lim ξ→0 ⟨λ 4 ⟩ = 2 + 8π 4 45 ξ 4 ; (IV.8) lim ξ→0 ⟨λ 6 ⟩ = 5 + 16π 4 15 ξ 4 . (IV.9) Numerically evaluating these expressions in the T > Tc regime as functions of ξ yields the results shown in [PIT… view at source ↗
Figure 3
Figure 3. Figure 3: For d = 2, the higher-order moments increase without bound as ξ increases. As a result, regardless of the existence of a phase transition, the moments of the spectrum continue to increase with the correlation length. In contrast, for d ≥ 3, the moments saturate to a constant value as ξ → ∞. Evaluating the relevant integrals in the ξ → ∞ limit, we obtain, for d = 3 and d = 4: lim ξ→∞ ⟨λ 4 ⟩ ≈ 3.16, lim ξ→∞ … view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Fourth and sixth moments of the eigenvalue spectrum of Ising Model (top) and Gaussian EA spin glass [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Scaling of the fourth and sixth moments of the eigenvalue spectrum at [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison between Monte Carlo data (blue) and self-consistent solution from the Vector Dyson Equation [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Spectral edge Σ = 2 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Spectral edge Σ = 2 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: A statistical physics model on a triangular grid [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

We introduce a random matrix framework for studying statistical-mechanical lattice systems through spectral observables. Equilibrium configurations sampled from a Boltzmann measure are mapped to matrix ensembles whose covariance structure is inherited from the spatial correlations of the underlying model. This construction maps real-space correlation functions to a momentum-space variance profile, providing a direct bridge between statistical-mechanical correlations and correlated random matrix ensembles. We derive this variance profile in finite-correlation-length and critical regimes, and compute spectral moments within a Wick-contraction expansion. A complementary self-consistent description of the bulk density is developed using the resolvent formalism. These analytical methods are benchmarked against Monte Carlo data for the two-dimensional Ising model and three-dimensional Edwards--Anderson spin glasses. In both cases, the spectra evolve from the semicircle law at high temperature to model-dependent critical forms reflecting the structure of correlations. The framework, therefore, provides a quantitative spectral route to probing collective behavior in ordered and disordered statistical systems, while also defining a class of physically motivated correlated random matrix ensembles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces a random matrix framework for statistical-mechanical lattice systems by mapping Boltzmann-sampled equilibrium configurations to matrix ensembles with covariance inherited from spatial correlations. It derives a variance profile in finite-correlation-length and critical regimes, computes spectral moments via Wick-contraction expansion, and develops a self-consistent resolvent description for the bulk density. These are benchmarked with Monte Carlo data for the 2D Ising and 3D Edwards-Anderson models, showing spectra evolving from the semicircle law at high temperature to model-dependent critical forms.

Significance. Should the central mapping and derivations prove robust, this framework offers a quantitative spectral method to probe collective behavior in ordered and disordered systems, while defining a new class of physically motivated correlated random matrix ensembles. The Monte Carlo benchmarks for concrete models constitute a strength, providing an external check on the theoretical predictions.

major comments (1)
  1. [Derivation of spectral moments] The computation of spectral moments employs a Wick-contraction expansion, which assumes that the random variables obey Wick's theorem with vanishing cumulants of order greater than 2. Given that the lattice configurations consist of discrete ±1 spins sampled from a non-Gaussian Boltzmann measure, and that local fluctuations are non-Gaussian at criticality, this assumption may not hold. Consequently, the derived moments could receive uncontrolled corrections, which would affect the validity of the analytic bridge between spatial correlations and the bulk spectrum.
minor comments (2)
  1. [Abstract and benchmarking section] The abstract and results section cite Monte Carlo benchmarks but omit error bars, quantitative fit metrics (e.g., R² or deviation measures), and exclusion criteria for the data. Including these would strengthen the verifiability of the agreement between theory and simulation.
  2. [Notation and presentation] Ensure that the definition of the variance profile and its mapping from real-space correlations to momentum-space is clearly distinguished from any fitted parameters to avoid potential circularity concerns.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting an important point regarding the assumptions underlying our spectral moment calculations. We address this comment below and will make corresponding revisions to clarify the scope and limitations of the approach.

read point-by-point responses

  1. Referee: The computation of spectral moments employs a Wick-contraction expansion, which assumes that the random variables obey Wick's theorem with vanishing cumulants of order greater than 2. Given that the lattice configurations consist of discrete ±1 spins sampled from a non-Gaussian Boltzmann measure, and that local fluctuations are non-Gaussian at criticality, this assumption may not hold. Consequently, the derived moments could receive uncontrolled corrections, which would affect the validity of the analytic bridge between spatial correlations and the bulk spectrum.

    Authors: We agree that the Wick-contraction expansion is an approximation that assumes vanishing higher-order cumulants, which does not strictly hold for discrete ±1 spins drawn from a non-Gaussian Boltzmann measure, particularly at criticality where fluctuations are known to be non-Gaussian. In the manuscript this expansion is used to obtain explicit, closed-form expressions for the low-order spectral moments directly from the momentum-space variance profile. While higher cumulants will in principle generate corrections, the Monte Carlo benchmarks for both the 2D Ising and 3D Edwards-Anderson models show quantitative agreement with the predicted moments and resolvent densities over a wide temperature range. This empirical support suggests that the leading Gaussian contributions dominate the bulk spectral features of interest. In the revised manuscript we will add an explicit discussion of this approximation, state its range of validity (including the high-temperature regime where fluctuations become effectively Gaussian), and outline possible routes for systematic inclusion of higher cumulants in future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines a mapping from Boltzmann-sampled lattice configurations to a correlated random-matrix ensemble by inheriting the covariance directly from the model's real-space correlation function, then applies standard RMT tools (Wick moment expansion and resolvent self-consistency) to obtain the spectrum. These steps are benchmarked against independent Monte Carlo data for the 2D Ising and 3D Edwards-Anderson models. No equation reduces the claimed spectral evolution to a tautological re-expression of the input correlations, no load-bearing premise rests on a self-citation chain, and the central bridge between spatial correlations and matrix spectra retains independent content beyond the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard Wick contractions for moments and a resolvent self-consistency equation whose inputs are the model's correlation functions; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Wick contraction expansion remains valid for the spectral moments of the constructed ensemble
    Invoked to compute moments from the variance profile.
  • domain assumption The resolvent formalism yields a self-consistent bulk density for the correlated ensemble
    Used to obtain the eigenvalue density beyond the moment expansion.

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