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 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 R, while the latter is in 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.INTERACTING BROWNIAN MOTIONS WITH LOG POTENTIALS 5 problem to prove the Markov property of their processes and identify these two processes. 2 We also refer to [5][6][7] and [19] for stochastic processes of one-dimensional infinite particle systems related to random matrices.As for two-dimensional infinite systems with logarithmic interactions, the construction of stochastic processes based on the explicit computation of space-time correlation functions has not been done. Techniques useful in one-dimension, such as applying the Karlin-McGregor formula, are no longer valid in two dimensions.Let us briefly explain the main idea. We introduce the notion of quasi-Gibbs measures as a substitution for Gibbs measures. These measures satisfy inequality (2.8) involving a (finite volume) Hamiltonian. Inequality (2.8) is sufficient for the closability of the Dirichlet forms and the construction of the diffusions.To obtain the above-mentioned inequality we control the difference of the infinite volume Hamiltonians instead of the Hamiltonian, itself. The key point of the control is the usage of the geometric property of the random point fields behind the dynamics. Indeed, although the difference still diverges for Poisson random fields and Gibbs measures with translation invariance, it becomes finite for random point fields such as Dyson random point fields and Ginibre random point fields. For these random point fields the fluctuations of particles are extremely suppressed because the logarithmic potentials are quite strong. This cancels the sum of the difference of the infinite-volume Hamiltonians.The organization of the paper is as follows. In Section 2, we describe the set-up and state the main results (Theorems 2.2 and 2.3). We first introduce the notion of quasi-Gibbs measures and give a general resul...

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