Local Statistics of Dyson Brownian Motion

Abstract

The time to equilibrium of the local statistics of Dyson Brownian motion with general initial data is investigated. In the bulk of the spectrum it is established that the local statistics coincide with the invariant GOE/GUE after a time $t$ if the density of states is bounded away from $0$ and above at all scales $\eta \gtrsim t$. At the edge of the spectrum, the statistics of the extremal particles coincide after time $t$ with those of the GOE/GUE if the initial data has a suitable square-root behavior down to scales $\eta \gtrsim t^2$. The approach is based on a reduction to a discrete parabolic equation. The local statistics of Dyson Brownian motion are studied by analysis of the solution to this parabolic equation.