Asymptotics of a discrete-time particle system near a reflecting boundary

TL;DR

This paper analyzes the asymptotic behavior of a discrete-time particle system with a reflecting boundary, revealing connections to determinantal processes and special kernels, and highlighting differences in multi-level dynamics.

Contribution

It establishes a link between the particle system and a Markov chain from orthogonal group representations, deriving asymptotics and kernels in the wall universality class.

Findings

Derived discrete Jacobi and symmetric Pearcey kernels in the asymptotic limit.

Showed the particle system is equivalent to a Markov chain from orthogonal group representations.

Identified differences in evolution between multi-level particle systems and associated Markov chains.

Abstract

We examine a discrete-time Markovian particle system on the quarter-plane introduced by M. Defosseux. The vertical boundary acts as a reflecting wall. The particle system lies in the Anisotropic Kardar-Parisi-Zhang with a wall universality class. After projecting to a single horizontal level, we take the longtime asymptotics and obtain the discrete Jacobi and symmetric Pearcey kernels. This is achieved by showing that the particle system is identical to a Markov chain arising from representations of the infinite-dimensional orthogonal group. The fixed-time marginals of this Markov chain are known to be determinantal point processes, allowing us to take the limit of the correlation kernel. We also give a simple example which shows that in the multi-level case, the particle system and the Markov chain evolve differently.