Interacting Infinite Particle Systems and Random Matrices

众所周知，Dyson布朗运动模型中粒子位置的分布与厄米特矩阵值过程的特征值的分布是一致的，其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，其中非碰撞条件是在有限时间区间(0，T]内施加的。这是一个时间上的非均匀扩散过程，在T到无穷大的范围内，它收敛于Dyson模型。我们构造了一个厄米矩阵值过程，使得它的特征值与我们的非碰撞布朗运动系统的粒子位置相同分布。作为对标准高斯随机矩阵系综的Dyson模型理论的推广，我们系统地研究了厄米矩阵值过程。除了非碰撞布朗运动外，我们还引入了广义曲折的非碰撞系统，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实现为粒子分布，然后我们证明了这些非碰撞扩散是Pfaffian过程，即任何多时间关联函数都由Pfaffian给出。在无限粒子极限下，我们证明了所得到的无限Pfaffian过程的矩阵核的元素一般由构成Bessel函数的函数的Riemann-Liouville积分来表示。

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.