Interacting Infinite Particle Systems and Random Matrices

已知Dyson的布朗运动模型中粒子位置的分布与Hermitian矩阵值过程的特征值的分布相一致，其中Hermitian矩阵值过程的项是独立的布朗运动。考虑了一类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。这是一个时间非齐次扩散过程，在T到无穷大的极限内，它收敛到戴森模型。本文构造了一个特征值与非碰撞布朗运动系统的粒子位置同分布的Hermitian矩阵值过程，作为标准高斯随机矩阵系综Dyson模型理论的推广，对Hermitian矩阵值过程进行了系统的研究。除了非碰撞的布朗运动外，我们还引入了广义曲折的非碰撞扩散系统，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可以实现为粒子分布，并证明了这些非碰撞扩散是Pfronan过程，即任何多时间相关函数都由Pfronan给出.在无穷粒子极限下，我们证明了所得到的无穷Pfronan过程的矩阵核的元素一般都由包含Bessel函数的函数的Riemann-Liouville微分积分表示.

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.