Phase transitions in the ASEP and stochastic six-vertex model

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 shape 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- Péché 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. Keywords: asymmetric simple exclusion process, stochastic six-vertex model, Baik–Ben–Arous–Péché phase transition, stochastic higher spin vertex models, Kardar–Parisi–Zhang universality class