The Annals of Probability

Phase transitions in the ASEP and stochastic six-vertex model

Amol Aggarwal and Alexei Borodin

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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
Received: August 2016
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

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

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

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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References

  • [1] Aggarwal, A. (2017). Convergence of the stochastic six-vertex model to the ASEP. Math. Phys. Anal. Geom. 20 Article 3.
  • [2] Aggarwal, A. (2018). Current fluctuations of the stationary ASEP and stochastic six-vertex model. Duke Math. J. 167 269–384.
  • [3] Aggarwal, A. and Borodin, A. (2019). Supplement to “Phase transitions in the ASEP and stochastic six-vertex model.” DOI:10.1214/17-AOP1253SUPP.
  • [4] Amir, G., Corwin, I. and Quastel, J. (2011). Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions. Comm. Pure Appl. Math. 64 466–537.
  • [5] Baik, J. (2006). Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 133 205–235.
  • [6] Baik, J., Ben-Arous, G. and Péché, S. (2005). Phase transition for the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697.
  • [7] Balázs, M. and Seppäläinen, T. (2009). Fluctuation bounds for the asymmetric simple exclusion process. ALEA Lat. Am. J. Probab. Math. Stat. 6 1–24.
  • [8] Balázs, M. and Seppäläinen, T. (2010). Order of current variance and diffusivity in the asymmetric simple exclusion process. Ann. of Math. (2) 171 1237–1265.
  • [9] Barraquand, G. (2015). A phase transition for $q$-TASEP with a few slower particles. Stochastic Process. Appl. 125 2674–2699.
  • [10] Baxter, R. J. (1989). Exactly Solved Models in Statistical Mechanics. Academic Press, London.
  • [11] Ben-Arous, G. and Corwin, I. (2011). Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture. Ann. Probab. 39 104–138.
  • [12] Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle system. Comm. Math. Phys. 183 571–607.
  • [13] Borodin, A. (2018). Stochastic higher spin six vertex models and Macdonald measures. J. Math. Phys. 59.
  • [14] Borodin, A. and Corwin, I. (2014). Macdonald processes. Probab. Theory Related Fields 158 225–400.
  • [15] Borodin, A., Corwin, I. and Ferrari, P. L. (2014). Free energy fluctuations for directed polymers in random media in $1+1$ dimension. Comm. Pure Appl. Math. 67 1129–1214.
  • [16] Borodin, A., Corwin, I. and Gorin, V. (2016). Stochastic six-vertex model. Duke Math. J. 165 563–624.
  • [17] Borodin, A., Corwin, I., Petrov, L. and Sasamoto, T. (2015). Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz. Comm. Math. Phys. 339 1167–1245.
  • [18] Borodin, A., Corwin, I. and Sasamoto, T. (2014). From duality to determinants for $q$-TASEP and ASEP. Ann. Probab. 42 2314–2382.
  • [19] Borodin, A. and Ferrari, P. L. (2008). Large time asymptotics of growth models on space-like paths. I. PushASEP. Electron. J. Probab. 13 1380–1418.
  • [20] Borodin, A. and Ferrari, P. L. (2014). Anisotropic growth of random surfaces in $2+1$ dimensions. Comm. Math. Phys. 325 603–684.
  • [21] Borodin, A. and Petrov, L. (2018). Higher spin six-vertex models and rational symmetric functions. Selecta Math. (N.S.) 24 751–874.
  • [22] Borodin, A. and Petrov, L. (2017). Lectures on integrable probability: Stochastic vertex models and symmetric functions. In Stochastic Processes and Random Matrices (G. Schehr, A. Altland, Y. V. Fyodorov, N. O’Connell and L. F. Cugliandolo, eds.). Lecture Notes of the les Houches Summer School 104 26–131. Oxford Univ. Press, London.
  • [23] Calabrese, P., Le Doussal, P. and Rosso, A. (2010). Free-energy distribution of the directed polymer at high temperature. EPL (Europhysics Letters) 90 20002.
  • [24] Corwin, I. (2012). The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1 1130001.
  • [25] Corwin, I. and Quastel, J. (2013). Crossover distributions at the edge of the rarefaction fan. Ann. Probab. 41 1243–1314.
  • [26] Dotsenko, V. (2010). Replica Bethe ansatz derivation of the Tracy–Widom distribution of the free energy fluctuations in one-dimensional directed polymers. J. Stat. Mech. Theory Exp. 2010 P07010.
  • [27] Gwa, L.-H. and Spohn, H. (1992). Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys. Rev. Lett. 68 725–728.
  • [28] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559–664.
  • [29] Hairer, M. (2014). A theory of regularity structures. Invent. Math. 198 269–504.
  • [30] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
  • [31] Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.
  • [32] Lee, E. (2010). Distribution of a particle’s position in the ASEP with the alternating initial condition. J. Stat. Phys. 140 635–647.
  • [33] Lieb, E. H. (1967). Residual entropy of square ice. Phys. Rev. Lett. 162 162–172.
  • [34] MacDonald, J., Gibbs, J. and Pipkin, A. (1968). Kinetics of biopolymerization on nucleic acid templates. Biopolymers 6 1–25.
  • [35] Okounkov, A. (2001). Infinite wedge and random partitions. Selecta Math. (N.S.) 7 57–81.
  • [36] Ortmann, J., Quastel, J. and Remenik, D. (2016). Exact formulas for random growth with half-flat initial data. Ann. Appl. Probab. 26 507–548.
  • [37] Ortmann, J., Quastel, J. and Remenik, D. (2017). A Pfaffian representation for flat ASEP. Comm. Pure Appl. Math. 70 3–89.
  • [38] Péché, S. (2006). The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields 134 127–173.
  • [39] Prähofer, M. and Spohn, H. (2002). Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium 51 (V. Sidoravicius, ed.) 185–204. Birkhäuser, Basel.
  • [40] Quastel, J. (2011). Introduction to KPZ. Current Developments in Mathematics 2011 125–194.
  • [41] Sasamoto, T. and Spohn, H. (2010). Exact height distribution for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834 523–542.
  • [42] Spitzer, F. (1970). Interaction of Markov processes. Adv. Math. 5 246–290.
  • [43] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the airy kernel. Comm. Math. Phys. 159 151–174.
  • [44] Tracy, C. A. and Widom, H. (2008). A Fredholm determinant representation in ASEP. J. Stat. Phys. 132 291–300.
  • [45] Tracy, C. A. and Widom, H. (2008). Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys. 279 815–844.
  • [46] Tracy, C. A. and Widom, H. (2009). Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290 129–154.
  • [47] Tracy, C. A. and Widom, H. (2009). On ASEP with step Bernoulli initial condition. J. Stat. Phys. 137 825–838.

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.