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Interacting Infinite Particle Systems and Random Matrices

无限粒子系统和随机矩阵的相互作用

基本信息

批准号：
15540106
负责人：
TANEMURA Hideki
金额：
$ 2.11万
依托单位：
CHIBA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540106/
关键词：
random matrix noncolliding process Brownian motion Bessel process generalized meander determinantal process Dyson process Airy process determinantal point process stochastic epidemic model duality coexistence Domany-Kinzel model

项目摘要

It is known that the distribution of particle positions in Dyson's model of Brownian motions coincides with that of eigenvalues of a Hermitian matrix-valued process, whose entries are independent Brownian motions. We considered a system of noncolliding Brownian motions, in which the noncolliding condition is imposed in a finite time interval (0,T]. This is a temporally inhomogeneous diffusion process and in the limit T to infinity, it converges to Dyson's model. We constructed such a Hermitian matrix-valued process that its eigenvalues are identically distributed with the particle positions of our system of noncolliding Brownian motions.As an extension of the theory of Dyson's models for the standard Gaussian random-matrix ensembles, we made a systematic study of hermitian matrix-valued processes. In addition to the noncolliding Brownian motions, we introduced noncolliding systems of generalized meanders and showed that all of the ten classes of eigenvalue statistics in the Altland-Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems.Then we proved that these non-colliding diffusions are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we showed that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann-Liouville differintegrals of functions comprising the Bessel functions.

已知戴森布朗运动模型中的粒子位置分布与厄米矩阵值过程的特征值分布一致，厄米矩阵值过程的项是独立的布朗运动。我们考虑了一个非碰撞布朗运动系统，其中非碰撞条件在有限时间间隔（0，T）内施加。这是一个时间上的非均匀扩散过程在极限T→∞时，它收敛于戴森模型。我们构造了这样一个厄米矩阵值过程，它的特征值与我们的非碰撞布朗运动系统的粒子位置分布相同。作为标准高斯随机矩阵系综的Dyson模型理论的扩展，我们对厄米矩阵值过程进行了系统的研究。除了非碰撞布朗运动外，我们还引入了广义弯曲的非碰撞系统，并证明了Altland-Zirnbauer分类中的所有十类特征值统计量在这些扩散粒子系统的特殊情况下都被实现为粒子分布。然后我们证明了这些非碰撞扩散是pfaffan过程，因为任何多时间相关函数都是由pfaffan给出的。在无限粒子极限下，我们证明了得到的无限Pfaffian过程的矩阵核的元素一般用包含贝塞尔函数的函数的Riemann-Liouville微分积分表示。