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January 2013 Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials
Hirofumi Osada
Ann. Probab. 41(1): 1-49 (January 2013). DOI: 10.1214/11-AOP736


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.


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Hirofumi Osada. "Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials." Ann. Probab. 41 (1) 1 - 49, January 2013.


Published: January 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1271.60105
MathSciNet: MR3059192
Digital Object Identifier: 10.1214/11-AOP736

Primary: 60J60 , 60K35
Secondary: 82B21 , 82C22

Keywords: Coulomb potentials , Diffusions , Dirichlet forms , Dyson’s model , Ginibre random point field , infinitely many particle systems , Interacting Brownian particles , logarithmic potentials , random matrices

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 1 • January 2013
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