Identifying Patterns Using Cross-Correlation Random Matrices Derived from Deterministic and Stochastic Differential Equations

TL;DR

This paper explores how the spectral properties of cross-correlation random matrices derived from differential equations can identify phase transitions and chaos in physical systems, potentially replacing more computationally intensive methods.

Contribution

It introduces a novel approach using spectral analysis of matrices from deterministic and stochastic differential equations to detect critical and chaotic behavior, bypassing traditional simulation techniques.

Findings

Spectral properties indicate phase transitions in spin systems.

Eigenvalue analysis captures chaotic dynamics.

Method reduces reliance on Monte Carlo simulations.

Abstract

Cross-Correlation random matrices have emerged as a promising indicator of phase transitions in spin systems. The core concept is that the evolution of magnetization encapsulates thermodynamic information [R. da Silva, Int. J. Mod. Phys. C, 2350061 (2023)], which is directly reflected in the eigenvalues of these matrices. When these evolutions are analyzed in the mean-field regime, an important question arises: Can the Langevin equation, when translated into maps, perform the same function? Some studies suggest that this method may also capture the chaotic behavior of certain systems. In this work, we propose that the spectral properties of random matrices constructed from maps derived from deterministic or stochastic differential equations can indicate the critical or chaotic behavior of such systems. For chaotic systems, we need only the evolution of iterated Hamiltonian equations, and for spin systems, the Langevin maps obtained from mean-field equations suffice, thus avoiding the need for Monte Carlo (MC) simulations or other techniques.