Scaling limit of the TASEP speed process

TL;DR

This paper demonstrates that the multi-type stationary distribution of TASEP converges to the stationary horizon (SH) process under scaling, linking TASEP, Busemann processes, and the KPZ universality class.

Contribution

It establishes the scaling limit of the TASEP speed process to the stationary horizon, providing evidence for universality in the KPZ class.

Findings

TASEP speed process scales to the stationary horizon around zero speed.

Connects SH with multiclass particle configurations.

Supports SH as a universal limit in KPZ-related models.

Abstract

We show that the multi-type stationary distribution of the totally asymmetric simple exclusion process (TASEP) scales to a nontrivial limit around the Bernoulli measure of density $1/2$. This is obtained by showing that the TASEP speed process, introduced by Amir, Angel and Valk\'o, scales around the speed $v=0$ to the stationary horizon (SH), a function-valued stochastic process recently introduced and studied by the authors, SH is believed to be the universal scaling limit of Busemann processes in the KPZ universality class. Our results add to the evidence for this universality by connecting SH with multiclass particle configurations. Previously SH has been associated with the exponential corner growth model, Brownian last-passage percolation, and the directed landscape.