Interacting particle systems at the edge of multilevel Jack processes

TL;DR

This paper investigates a multilevel particle system derived from Jack symmetric functions, analyzing its asymptotic behavior at the edge and revealing a limit described by a zero range process with local interactions.

Contribution

It introduces a new multilevel Markov chain model based on Jack functions and characterizes its edge asymptotics, connecting it to zero range processes.

Findings

Asymptotic separation of particles at the edge is established.

The limit process is identified as a zero range process with local interactions.

The model provides a discretization of multilevel Dyson Brownian motion.

Abstract

We consider a multilevel continuous time Markov chain $X(s;N) = (X_i^j(s;N): 1 \leq i \leq j \leq N)$, which is defined by means of Jack symmetric functions and forms a certain discretization of the multilevel Dyson Brownian motion. The process $X(s;N)$ describes the evolution of a discrete interlacing particle system with push-block interactions between the particles, which preserve the interlacing property. We study the joint asymptotic separation of the particles at the right edge of the ensemble as the number of levels and time tend to infinity and show that the limit is described by a certain zero range process with local interactions.