Markov processes of infinitely many nonintersecting random walks
type: Item
instance of: scholarly article
title: Markov processes of infinitely many nonintersecting random walks
zbMATH Open document ID: 1278.60113
author: Alexei Borodin
author: Vadim Gorin
publication date: 2013-05-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.1299
review text: Let \(X(t)=(X_1(t),\dotsc, X_N(t))\in {\mathbb Z}^N\) be \(N\geq 1\) independent rate one Poisson processes started at \(X(0)=(0, 1, \dotsc, N-1)\) and conditioned to finish at \(X(T)=Y\) while taking mutually distinct values for all \(0\leq t\leq T\). Thus, \(\{X_i(t)\}_{i\geq 1}\) form \(N\) nonintersecting paths in \({\mathbb Z}\times [0, T]\). A result of [\textit{W. König} et al., Electron. J. Probab. 7, Paper No. 5 (2002; Zbl 1007.60075)] says that as \(T\to \infty\) with \(Y\) being asymptotically linear, \(Y\sim T\xi\) with a collection of asymptotic speeds \(\xi=(\xi_1\leq \dotsb\leq \xi_N) \in{\mathbb R}_{>0}^N\), the process \((X(t), 0\leq t\leq T)\) has a limit \((\chi(t), t\geq 0)\), which is a homogeneous Markov process on \({\mathbb Z}^N\) with initial condition \(X(0)\) and transition probabilities over time interval \(t\) given by \(\mathrm{const} {{\Xi_N(y)}\over{\Xi_N(x)}}\det_{i,j=1,\dotsc, N}\left[{{t^{y_i - x_j}}\over {(y_i - x_j)!}}\right]\), where \(\Xi_N(u)=\det\left[\xi_i^{u_j}\right]_{i, j=1}^N\). The process \(\Xi(t)\) is the Doob \(h\)-transform of \(N\) independent Poisson processes with respect to the harmonic function \(h=\Xi_N\). In the case of equal speeds, the distribution of \(X(t)\) is known as the Charlier orthogonal polynomial ensemble, which is the basic discrete probabilistic model of random matrix type. The procedure of passing to a limit by increasing the number of particles and scaling the space appropriately in models of random matrix type is very well developed. In many cases, such a limit can also be successfully performed for joint distributions at finitely many times if the original model undergoes a Markov dynamics. The most common approach is to control the limiting behavior of local correlation functions. It is much less clear what happens to the Markovian structure of the dynamics under such limit transitions. One goal of the paper is to prove that, if \(\xi_j\)'s form a geometric progression with ratio \(q^{-1}>1…
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60G55
zbMATH DE Number: 6162388
zbMATH Keywords: non-intersecting paths
zbMATH Keywords: infinite-dimensional Markov process
zbMATH Keywords: determinantal point process
zbMATH Keywords: Gelfand-Tsetlin scheme
reviewed by: Viktor Ohanyan
MaRDI profile type: Publication
OpenAlex ID: W2096307945
arXiv ID: 1106.1299
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DOI: 10.1007/S00440-012-0417-4
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article