Random matrices theory elucidates the critical nonequilibrium phenomena

TL;DR

This paper applies random matrix theory to analyze the eigenvalue distribution of correlation matrices in a spin system, revealing phase transition signatures that could be useful for characterizing critical phenomena.

Contribution

It introduces a novel approach using magnetization correlation matrices and eigenvalue distributions to detect phase transitions in spin systems.

Findings

Eigenvalue distribution transitions at critical temperature

Eigenvalue gap indicates phase change

Marchenko-Pastur law applies in paramagnetic phase

Abstract

The earlier times of evolution of a magnetic system contain more information than we can imagine. Capturing correlation matrices G of different time evolutions of a simple testbed spin system, as the Ising model, we analyzed the density of eigenvalues of G^{T}G for different temperatures. We observe a transition of the shape of the distribution that presents a gap of eigenvalues from critical temperature with a continuous migration to the Marchenko-Pastur law for the paramagnetic phase. We consider the analysis a promising method to be applied in other spin systems to characterize phase transitions. Our approach is different from alternatives in the literature since it uses the magnetization matrix and not the spatial matrix of spins.