Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials

Abstract

We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in $\mathbb{R}^d$ and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures. We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Brownian motions. The former is a particle system in $\mathbb{R}$, while the latter is in $\mathbb{R}^2$. Both models are translation and rotation invariant in space, and as such, are prototypes of dimensions $d=1,2$, respectively. The equilibrium states of the former diffusion model are determinantal or Pfaffian random point fields with sine kernels. They appear in the thermodynamical limits of the spectrum of the ensembles of Gaussian random matrices such as GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble.