Nonintersecting Paths, Noncolliding Diffusion Processes and Representation Theory

摘要

The system of one-dimensional symmetric simple random walks, in which none ofwalkers have met others in a given time period, is called the vicious walkermodel. It was introduced by Michael Fisher and applications of the model tovarious wetting and melting phenomena were described in his Boltzmann medallecture. In the present report, we explain interesting connections amongrepresentation theory, probability theory, and random matrix theory using thissimple diffusion particle system. Each vicious walk of $N$ walkers isrepresented by an $N$-tuple of nonintersecting lattice paths on thespatio-temporal plane. There is established a simple bijection betweennonintersecting lattice paths and semistandard Young tableaux. Based on thisbijection and some knowledge of symmetric polynomials called the Schurfunctions, we can give a determinantal expression to the partition function ofvicious walks, which is regarded as a special case of the Karlin-McGregorformula in the probability theory (or the Lindstr\"om-Gessel-Viennot formula inthe enumerative combinatorics). Due to a basic property of Schur function, wecan take the diffusion scaling limit of the vicious walks and define anoncolliding system of Brownian particles. This diffusion process solves thestochastic differential equations with the drift terms acting as the repulsivetwo-body forces proportional to the inverse of distances between particles, andthus it is identified with Dyson's Brownian motion model. In other words, theobtained noncolliding system of Brownian particles is equivalent indistribution with the eigenvalue process of a Hermitian matrix-valued process.