Random Matrix Spectra from Boltzmann-Weighted Lattice Ensembles

TL;DR

This paper develops a spectral analysis framework linking statistical-mechanical lattice models to correlated random matrix ensembles, enabling insights into phase transitions and collective behaviors.

Contribution

It introduces a novel random matrix approach that maps lattice correlations to spectral properties, bridging statistical mechanics and random matrix theory.

Findings

Spectra transition from semicircle law at high temperature to model-dependent forms at criticality.

Analytical spectral moments match Monte Carlo data for Ising and spin glass models.

Framework provides a quantitative spectral method to study collective phenomena in complex systems.

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.