Interacting Brownian particles and the Wigner law

Abstract

In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX j (t)=δ n dB j (t) −D jOn(X 1,...,X n)dt,j=1, 2,...,n. Here theB jare independent Brownian motions in ℝ d , and O n (X 1,...,X n)=α n ∑∑ i≠j V(X i∔X j) + θn∑i U(X 1). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=−log|x|,U(x)=x 2/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast→∞. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.