Consider n nonintersecting Brownian motions on ℝ, which leave from p definite points and are forced to end up at q points at time t = 1. When n → ∞, the equilibrium measure for these Brownian particles has its support on p intervals, for t ∽ 0, and on q intervals, for t ∽ 1. Hence it is clear that, when t evolves, intervals must merge, must disappear and be created, leading to various phase transitions between times t = 0 and 1.

Near these moments of phase transitions, there appears an infinite-dimensional diffusion, a Markov cloud, in the limit n ↗ ∞, which one expects to depend only on the nature of the singularity associated with this phase change. The transition probabilities for these Markov clouds satisfy nonlinear PDE’s, which are obtained from taking limits of the Brownian motion model with finite particles; the finite model is closely related to Hermitian matrix integrals, which themselves satisfy nonlinear PDE’s. The latter are obtained from investigating the connection between the Karlin-McGregor formula, moment matrices, the theory of orthogonal polynomials and the associated integrable systems. Various special cases are provided to illustrate these general ideas. This is based on work by Adler and van Moerbeke.

1. Introduction

This lecture in honor of Henry McKean forms a step in the direction of understanding the behavior of nonintersecting Brownian motions on ℝ (Dyson's Brownian motions), when the number of particles tends to ∞. It explains a novel interface between diffusion theory, integrable systems and the theory of orthogonal polynomials. These subjects have been at the center of Henry McKean's oeuvre.