Determinantal Martingales and Correlations of Noncolliding Random Walks

Abstract

We study the noncolliding random walk (RW), which is a particle system of onedimensional, simple and symmetric RWs starting from distinct even sites and conditioned never to collide with each other. When the number of particles is finite, N < ∞, this discrete process is constructed as an h-transform of absorbing RW in the N -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 h-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 N < ∞, 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 infiniteparticle systems, we obtain a family of equilibrium processes parameterized by particle density, which are determinantal with the discrete analogues of the extended sinekernel of Dyson's Brownian motion model with β = 2. Following Donsker's invariance principle, convergence of noncolliding RWs to the Dyson model is also discussed.

Determinantal martingales and noncolliding diffusion processes

“…Note that ℜZ(t) = V (t) ∈ Z and ℑZ(t) = W (t) ∈ R. The above results are summarized as follows [35].…”

“…Since we consider the noncolliding RW as a process represented by an unlabeled configuration (1.15), measurable functions of Ξ(·) are only symmetric functions of N variables, X j (·), 1 ≤ j ≤ N. Then by the equality (6.16), we obtain the following DMR and CPR for the present noncolliding RW [35].…”

Elliptic determinantal process of type A

“…The noncolliding diffusion processes have attracted much attention in probability theory also by the fact that they are realized as h-transforms in the sense of Doob of absorbing particle systems in the Weyl chambers [18,29,25]. The relationship between the above mentioned integrability as spatio-temporal models and h-transform constructions as stochastic processes has been clarified by introducing a notion of determinantal martingales in [28,23,24]. The purpose of the present paper is to report elliptic extensions of these determinantal processes.…”

Noncolliding system of continuous-time random walks

“…Moreover, we show a relaxation phenomenon to the equilibrium determinantal point process, which is governed by the sine kernel defined on Z [10].…”

“…For this purpose Corollary 2.2 and Theorem 2.4 in [16] proved by König, O'Connell, and Roch are useful. See also [15,4,10]. Here we rewrite their theorems with modifications to fit the present situation and put the following proposition.…”