The Annals of Probability

Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials

Hirofumi Osada

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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.

Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 1-49.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1358951980

Digital Object Identifier
doi:10.1214/11-AOP736

Mathematical Reviews number (MathSciNet)
MR3059192

Zentralblatt MATH identifier
1271.60105

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65]
Secondary: 82C22: Interacting particle systems [See also 60K35] 82B21: Continuum models (systems of particles, etc.)

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

Citation

Osada, Hirofumi. Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. Ann. Probab. 41 (2013), no. 1, 1--49. doi:10.1214/11-AOP736. https://projecteuclid.org/euclid.aop/1358951980


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