Universal exit probabilities in the TASEP

TL;DR

This paper derives universal exit probabilities for particles in the TASEP model, extending previous results to more general boundary conditions and demonstrating convergence to the Airy$_2$ process in the scaling limit.

Contribution

It introduces a method to compute exit probabilities for TASEP with staircase-like boundaries without ordering constraints, generalizing previous correlation function results.

Findings

Exit probabilities expressed as Fredholm determinants on boundary sets.

Removal of time ordering constraints in the analysis.

Convergence to the universal Airy$_2$ process in the scaling limit.

Abstract

We study the joint exit probabilities of particles in the totally asymmetric simple exclusion process (TASEP) from space-time sets of given form. We extend previous results on the space-time correlation functions of the TASEP, which correspond to exits from the sets bounded by straight vertical or horizontal lines. In particular, our approach allows us to remove ordering of time moments used in previous studies so that only a natural space-like ordering of particle coordinates remains. We consider sequences of general staircase-like boundaries going from the northeast to southwest in the space-time plane. The exit probabilities from the given sets are derived in the form of Fredholm determinant defined on the boundaries of the sets. In the scaling limit, the staircase-like boundaries are treated as approximations of continuous differentiable curves. The exit probabilities with respect to points of these curves belonging to arbitrary space-like path are shown to converge to the universal Airy$_2$ process.