Long-Range Correlated Random Matrices

TL;DR

This paper investigates how algebraic correlations in random matrices, modeled via a long-range correlated percolation, influence eigenvalue statistics and spectral density, revealing phase transitions and new spectral regimes.

Contribution

It introduces a model with power-law decaying correlations in matrix elements and analyzes their impact on spectral properties, uncovering a transition at a critical correlation exponent.

Findings

Eigenvalue distribution changes qualitatively with correlation decay exponent H.

At H_c=3/4, a transition to Gaussian eigenvalue statistics occurs.

Spectral density approaches the semicircle law for large H.

Abstract

Motivated by the importance ascribed to correlations in random matrices used to model phenomena in various scientific disciplines, we report how algebraic correlations between matrix elements affect the eigenvalue statistics and spectral density of random matrices. These correlations, introduced through a long-range correlated percolation model, decay as a power law $\propto r^{-2H}$, with exponent $H > 0$. As $H$ varies, both the eigenvalue distribution and excess kurtosis undergo qualitative changes. At the threshold $H_c = 3/4$, characterized by emergent Gaussian statistics, a sign change in excess kurtosis marks a transition from a fat-tailed generalized $t$-distribution to one that gradually approaches the standard semicircle law for $H \gg H_c$. Our analytical results, based on scaling analysis and supported by extensive numerical simulations, provide clear predictions and uncover novel spectral regimes in random matrix theory. Our results connect techniques from statistical physics, percolation theory, and random matrix analysis, offering a new perspective on universality in correlated ensembles.