Random transition-rate matrices for the master equation

TL;DR

This paper applies random-matrix theory to analyze the eigenvalue distributions of transition-rate matrices in the master equation, revealing distinct behaviors for symmetric and asymmetric cases relevant to stochastic system dynamics.

Contribution

It introduces new random matrix ensembles for transition-rate matrices and characterizes their eigenvalue distributions and correlations, highlighting differences from standard ensembles.

Findings

Eigenvalue distributions differ between symmetric and asymmetric matrices.

The fraction of real eigenvalues scales anomalously with matrix size.

Eigenvalue correlations depend on the symmetry of the transition rates.

Abstract

Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of independent rates of forward and backward transitions are considered. The first case leads to symmetric transition-rate matrices, whereas the second corresponds to general, asymmetric matrices. The resulting matrix ensembles are different from the standard ensembles and show different eigenvalue distributions. For example, the fraction of real eigenvalues scales anomalously with matrix dimension in the asymmetric case.