Scaling limit of the TASEP speed process

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 Valkó, 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.

The directed landscape

“…Busemann functions in the directed landscape were first studied in [RV21]. The law of the whole stationary horizon found in [Bus21,SS23b], connected to the directed landscape in [BSS24], and shown to be the scaling limit for stationary measures in discrete models in [BSS22,BSS23].…”