A Brownian-Motion Model for the Eigenvalues of a Random Matrix

TLDR

The authors introduce a Coulomb gas of n point charges undergoing Brownian motion with mutual electrostatic repulsion, whose dynamics—tuned by initial conditions—generate an ensemble of random matrices that model the Hamiltonian of a complex system with approximate conservation laws and capture the time evolution of eigenvalue statistics as conservation‑destroying interactions strengthen. They prove that this Brownian‑motion Coulomb gas exactly describes the eigenvalue behavior of an n×n Hermitian matrix with independently moving entries, establish a virial theorem for the gas, and derive several properties of its stationary state.

Abstract

A new type of Coulomb gas is defined, consisting of n point charges executing Brownian motions under the influence of their mutual electrostatic repulsions. It is proved that this gas gives an exact mathematical description of the behavior of the eigenvalues of an (n × n) Hermitian matrix, when the elements of the matrix execute independent Brownian motions without mutual interaction. By a suitable choice of initial conditions, the Brownian motion leads to an ensemble of random matrices which is a good statistical model for the Hamiltonian of a complex system possessing approximate conservation laws. The development with time of the Coulomb gas represents the statistical behavior of the eigenvalues of a complex system as the strength of conservation-destroying interactions is gradually increased. A ``virial theorem'' is proved for the Brownian-motion gas, and various properties of the stationary Coulomb gas are deduced as corollaries.