Interacting Infinite Particle Systems and Random Matrices

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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.