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Random matrix spectra from Boltzmann-weighted lattice ensembles

Yaprak Önder1,2,*, Abbas Ali Saberi3,2,†, and Roderich Moessner2

  • *Contact author: yaprakonder@g.harvard.edu
  • †Contact author: asaberi@constructor.university
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Phys. Rev. B 114, 204203 – Published 16 October, 2026

DOI: https://doi.org/10.1103/5xm2-81c1

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.

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Phys. Rev. B 114, 204203– Published 16 October, 2026
  • Received 19 May 2026
  • Revised 24 September 2026
  • Accepted 29 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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