Correlations for the Dyson Brownian motion model with Poisson initial conditions

TL;DR

This paper derives exact formulas for density-density correlations in the Dyson Brownian motion model with Poisson initial conditions, connecting random matrix eigenvalues and particle dynamics on a circle.

Contribution

It introduces a novel application of Jack polynomial theory to obtain explicit correlation expressions for the Dyson Brownian motion with Poisson initial states.

Findings

Exact correlation formulas for one and two particles at different times

Comparison of theoretical correlations with empirical eigenvalue data

Application to fluctuation analysis of linear statistics

Abstract

The circular Dyson Brownian motion model refers to the stochastic dynamics of the log-gas on a circle. It also specifies the eigenvalues of certain parameter-dependent ensembles of unitary random matrices. This model is considered with the initial condition that the particles are non-interacting (Poisson statistics). Jack polynomial theory is used to derive a simple exact expression for the density-density correlation with the position of one particle specified in the initial state, and the position of one particle specified at time $\tau$, valid for all $\beta > 0$. The same correlation with two particles specified in the initial state is also derived exactly, and some special cases of the theoretical correlations are illustrated by comparison with the empirical correlations calculated from the eigenvalues of certain parameter-dependent Gaussian random matrices. Application to fluctuation formulas for time displaced linear statistics in made.