Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials
Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials
复制标题
DOI：
10.1214/11-aop736
复制
发表时间：
2009-02
影响因子：
2.3
通讯作者：
H. Osada
中科院分区：
文献类型：
--
作者：
H. Osada
我们研究了无限多个布朗粒子通过对数函数(二维库仑势)在RD中运动并相互作用的扩散的结构。这些势能非常强，在自然界中的作用范围很长。相关的平衡态不再是吉布斯度量。我们给出了构造这种扩散的一般结果，并作为应用，构造了两个典型的具有对数相互作用势的相互作用布朗运动，即无限维Dyson模型和Ginibre相互作用布朗运动。前者是R中的质点系统，后者是R2中的质点系统。这两个模型在空间中都是平移和旋转不变的，因此分别是维度d=1，2的原型。前一种扩散模型的平衡态是正弦核的行列式或Pfaffian随机点场。它们出现在高斯随机矩阵系综谱的热力学极限中，如GOE、GUE和GSE。后一种扩散模型的平衡态是复数非厄米-高斯随机矩阵系综谱的热力学极限。
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.