The theory of random matrices plays an important role in many areas of pure mathematics
and employs a variety of sophisticated mathematical tools (analytical, probabilistic and …

We define a sequence of mappings \Gamma_k:D_0(R_+)^k→D_0(R_+)^k and prove the
following result: Let N_1,...,N_n be the counting functions of independent Poisson processes …

We survey a number of models from physics, statistical mechanics, probability theory and
combinatorics, which are each described in terms of an orthogonal polynomial ensemble …

As an extension of the theory of Dyson's Brownian motion models for the standard Gaussian
random-matrix ensembles, we report a systematic study of Hermitian matrix-valued …

We show that the function h(x)=i\ltj(x_j-x_i) is harmonic for any random walk in R^k with
exchangeable increments, provided the required moments exist. For the subclass of random …

The Airy $ _\beta $ line ensemble is a random collection of continuous curves, which should
serve as a universal edge scaling limit in problems related to eigenvalues of random …

We study a model of n non-intersecting squared Bessel processes in the confluent case: all
paths start at time t= 0 at the same positive value x= a, remain positive, and are conditioned …

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with
each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of …

We study systems of stochastic differential equations describing positions x_1,...,x_p of p
ordered particles, with inter-particles repulsions of the form H_ij(x_i,x_j)/(x_i-x_j). We show …

This book is based on my graduate-course lectures given at the Graduate School of
Mathematics of the University of Tokyo in October 2008 (at the invitation of T. Funaki and M …