### 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.

Original language | English |
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Pages (from-to) | 1-49 |

Number of pages | 49 |

Journal | Annals of Probability |

Volume | 41 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials.** / Osada, Hirofumi.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 41, no. 1, pp. 1-49. https://doi.org/10.1214/11-AOP736

}

TY - JOUR

T1 - Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials

AU - Osada, Hirofumi

PY - 2013/1/1

Y1 - 2013/1/1

N2 - We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in Rd 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 R2. 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.

AB - We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in Rd 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 R2. 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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UR - http://www.scopus.com/inward/citedby.url?scp=84874954001&partnerID=8YFLogxK

U2 - 10.1214/11-AOP736

DO - 10.1214/11-AOP736

M3 - Article

AN - SCOPUS:84874954001

VL - 41

SP - 1

EP - 49

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1

ER -