Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point

TL;DR

This paper develops a mathematical framework for analyzing a discrete-time TASEP with inhomogeneous rates, expressing its distribution via non-intersecting paths and Fredholm determinants, extending previous homogeneous models.

Contribution

It introduces a novel representation of the TASEP with inhomogeneous rates using non-intersecting paths and boundary-value problems, generalizing existing homogeneous rate results.

Findings

Transition kernel expressed via non-intersecting lattice paths

Joint distribution as a Fredholm determinant with a new kernel

Generalization to inhomogeneous rates with a finer structure

Abstract

We consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters. We use the combinatorics of the Robinson-Schensted-Knuth correspondence and certain intertwining relations to express the transition kernel of this interacting particle system in terms of ensembles of weighted, non-intersecting lattice paths and, consequently, as a marginal of a determinantal point process. We next express the joint distribution of the particle positions as a Fredholm determinant, whose correlation kernel is given in terms of a boundary-value problem for a discrete heat equation. The solution to such a problem finally leads us to a representation of the correlation kernel in terms of random walk hitting probabilities, generalising the formulation of Matetski, Quastel and Remenik (Acta Math., 2021) to the case of both particle- and time-inhomogeneous rates. The solution to the boundary value problem in the fully inhomogeneous case appears with a finer structure than in the homogeneous case.