Random Matrix Spectra from Boltzmann-Weighted Lattice Ensembles
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.
Versions
Related
- A multiscale lattice Boltzmann model for simulating Stokes to pre-Darcy flow
- NUMERICAL ANALYSIS OF A LAX-WENDROFF TYPE LATTICE BOLTZMANN SCHEME FOR THE ADVECTION EQUATION
- Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles
- Joint End-to-End Training of Weighted BP Decoder Ensembles
- Vectorial lattice Boltzmann solver for compressible inviscid flows with generic equation of state
BibTeX
@misc{onder2026random,
title = {{Random Matrix Spectra from Boltzmann-Weighted Lattice Ensembles}},
author = {Yaprak Önder and Abbas Ali Saberi and Roderich Moessner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.21254},
eprint = {2605.21254},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.21254},
}
RIS
TY - UNPB
TI - Random Matrix Spectra from Boltzmann-Weighted Lattice Ensembles
AU - Yaprak Önder
AU - Abbas Ali Saberi
AU - Roderich Moessner
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2605.21254
UR - https://arxiv.org/abs/2605.21254
AB - 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.
ER -