The Annals of Probability

Phase transitions in the ASEP and stochastic six-vertex model

Amol Aggarwal and Alexei Borodin

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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

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

Received: August 2016
Revised: September 2017
First available in Project Euclid: 26 February 2019

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]

Asymmetric simple exclusion process stochastic six-vertex model Baik–Ben–Arous–Péché phase transition stochastic higher spin vertex models Kardar–Parisi–Zhang universality class


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.

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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.