Phase Transitions in the ASEP and Stochastic Six-Vertex Model

TL;DR

This paper investigates phase transitions in current fluctuations of the ASEP and stochastic six-vertex models within the KPZ class, introducing a new initial data class and revealing a change in fluctuation behavior along a characteristic line.

Contribution

It introduces generalized step Bernoulli initial data for both models and demonstrates a phase transition in fluctuation exponents, linking to Baik-Ben-Arous-Péché distributions, a novel result for these models.

Findings

Identified a phase transition in fluctuation exponents from 1/2 to 1/3.

Established convergence to Baik-Ben-Arous-Péché distributions on the characteristic line.

Extended known results to stochastic six-vertex model for k > 1.

Abstract

In this paper we consider two models in the Kardar-Parisi-Zhang (KPZ) universality class, the asymmetric simple exclusion process (ASEP) and the stochastic six-vertex model. We introduce a new class of initial data (which we call {\itshape generalized step Bernoulli initial data}) for both of these models that generalizes the step Bernoulli initial data studied in a number of recent works on the ASEP. Under this class of initial data, we analyze the current fluctuations of both the ASEP and stochastic six-vertex model and establish the existence of a phase transition along a characteristic line, across which the fluctuation exponent changes from $1 / 2$ to $1 / 3$. On the characteristic line, the current fluctuations converge to the general (rank $k$) Baik-Ben-Arous-P\'{e}ch\'{e} distribution for the law of the largest eigenvalue of a critically spiked covariance matrix. For $k = 1$, this was established for the ASEP by Tracy and Widom; for $k > 1$ (and also $k = 1$, for the stochastic six-vertex model), the appearance of these distributions in both models is new.