The Annals of Probability

Universality of local statistics for noncolliding random walks

Vadim Gorin and Leonid Petrov

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Abstract

We consider the $N$-particle noncolliding Bernoulli random walk—a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some location $x$ (i.e., $x$ is in the “bulk”), and the initial configuration is balanced in a certain sense, then the space-time local statistics at $x$ are asymptotically governed by the extended discrete sine process (which can be identified with a translation invariant ergodic Gibbs measure on lozenge tilings of the plane). We also establish similar results for certain types of random initial data. Our proofs are based on a detailed analysis of the determinantal correlation kernel for the noncolliding Bernoulli random walk.

The noncolliding Bernoulli random walk is a discrete analogue of the $\boldsymbol{\beta}=2$ Dyson Brownian motion whose local statistics are universality governed by the continuous sine process. Our results parallel the ones in the continuous case. In addition, we naturally include situations with inhomogeneous local particle density on scale $T$, which nontrivially affects parameters of the limiting extended sine process, and in a particular case leads to a new behavior.

Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 2686-2753.

Dates
Received: September 2016
Revised: May 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1571731433

Digital Object Identifier
doi:10.1214/18-AOP1315

Mathematical Reviews number (MathSciNet)
MR4021235

Zentralblatt MATH identifier
07145301

Subjects
Primary: 60G55: Point processes
Secondary: 60C05: Combinatorial probability 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Noncolliding random walks determinantal point processes bulk universality discrete sine process steepest descent method Dyson’s conjecture homogenization

Citation

Gorin, Vadim; Petrov, Leonid. Universality of local statistics for noncolliding random walks. Ann. Probab. 47 (2019), no. 5, 2686--2753. doi:10.1214/18-AOP1315. https://projecteuclid.org/euclid.aop/1571731433


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