Determinantal Martingales and Correlations of Noncolliding Random Walks

摘要

We study the noncolliding random walk (RW), which is a particle system of one-dimensional, simple and symmetric RWs starting from distinct even sites and conditioned never to collide with each other. When the number of particles is finite, , this discrete process is constructed as an -transform of absorbing RW in the -dimensional Weyl chamber. We consider Fujita's polynomial martingales of RW with time-dependent coefficients and express them by introducing a complex Markov process. It is a complexification of RW, in which independent increments of its imaginary part are in the hyperbolic secant distribution, and it gives a discrete-time conformal martingale. The -transform is represented by a determinant of the matrix, whose entries are all polynomial martingales. From this determinantal-martingale representation (DMR) of the process, we prove that the noncolliding RW is determinantal for any initial configuration with , and determine the correlation kernel as a function of initial configuration. We show that noncolliding RWs started at infinite-particle configurations having equidistant spacing are well-defined as determinantal processes and give DMRs for them. Tracing the relaxation phenomena shown by these infinite-particle systems, we obtain a family of equilibrium processes parameterized by particle density, which are determinantal with the discrete analogues of the extended sine-kernel of Dyson's Brownian motion model with . Following Donsker's invariance principle, convergence of noncolliding RWs to the Dyson model is also discussed.