## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 47, Number 2 (2019), 613-689.

Dates
Revised: September 2017
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1551171634

Digital Object Identifier
doi:10.1214/17-AOP1253

Mathematical Reviews number (MathSciNet)
MR3916931

Zentralblatt MATH identifier
07053553

#### Citation

Aggarwal, Amol; Borodin, Alexei. Phase transitions in the ASEP and stochastic six-vertex model. Ann. Probab. 47 (2019), no. 2, 613--689. doi:10.1214/17-AOP1253. https://projecteuclid.org/euclid.aop/1551171634

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#### Supplemental materials

• Supplement to “Phase transitions in the ASEP and stochastic six-vertex model.”. This supplement serves as the Appendix for the present paper. In Appendix A, we provide some results about Fredholm determinants that are used in the asymptotic analysis above, and in Appendix B we outline an alternative way to establish Theorem 1.7 through a comparison with Schur measures.