Abstract
The continuous-time random walk is defined as a Poissonization of discrete-time random walk. We study the noncolliding system of continuous-time simple and symmetric random walks on . We show that the system is determinantal for any finite initial configuration without multiple point. The spatio-temporal correlation kernel is expressed by using the modified Bessel functions. We extend the system to the noncolliding process with an infinite number of particles, when the initial configuration has equidistant spacing of particles, and show a relaxation phenomenon to the equilibrium determinantal point process with the sine kernel.
Highlights
Eigenvalue distributions of Hermitian random-matrix ensembles provide typical examples of determinantal point processes (DPPs) on R [5,19]
The purpose of the present paper is to introduce a discrete model defined on a lattice Z, which realizes a determinantal process
We show that the system has a determinantal martingale representation
Summary
Eigenvalue distributions of Hermitian random-matrix ensembles provide typical examples of determinantal point processes (DPPs) on R [5,19]. We consider a continuous-time simple and symmetric random walk on Z, which is denoted by V (t), t ∈ [0, ∞) It is defined as a compound Poisson process such that its characteristic function ψV(t)(z) is given by [24], ψV (t)(z) := E eizV (t) = ∞ e−t t j (σ (z)) j j=0 j! The present continuous-time random walk is a Poissonization of discrete-time simple and symmetric random walk In this paper, this process on Z is denoted by RW.
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