Universality of local statistics for noncolliding random walks

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.