Identifying patterns using cross-correlation random matrices derived from deterministic and stochastic differential equations

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 34, 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.