The universal Airy_1 and Airy_2 processes in the Totally Asymmetric Simple Exclusion Process

TL;DR

This paper discusses the emergence of the universal Airy_1 and Airy_2 processes in the large-time limit of TASEP, linking them to growth models, last passage percolation, and random matrices.

Contribution

It introduces the definitions of Airy_1 and Airy_2 processes within TASEP and explains their universal nature and geometric representations connecting to other models.

Findings

Airy_2 process is universal across multiple models

Connections established between TASEP, growth models, and percolation

Geometric representation clarifies process origins

Abstract

In the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy_1 and Airy_2 processes. The Airy_2 process is an universal limit process occurring also in other models: in a stochastic growth model on 1+1-dimensions, 2d last passage percolation, equilibrium crystals, and in random matrix diffusion. The Airy_1 and Airy_2 processes are defined and discussed in the context of the TASEP. We also explain a geometric representation of the TASEP from which the connection to growth models and directed last passage percolation is immediate.