Duke Mathematical Journal

Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

Guillaume Barraquand, Alexei Borodin, Ivan Corwin, and Michael Wheeler

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time τ) according to the Tracy–Widom Gaussian orthogonal ensemble distribution on the τ1/3-scale. This is the first example of Kardar–Parisi–Zhang asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models.

Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall–Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity.

Article information

Duke Math. J., Volume 167, Number 13 (2018), 2457-2529.

Received: 12 May 2017
Revised: 16 April 2018
First available in Project Euclid: 27 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B23: Exactly solvable models; Bethe ansatz 05E05: Symmetric functions and generalizations 60H15: Stochastic partial differential equations [See also 35R60] 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Kardar–Parisi–Zhang universality class interacting particle systems asymmetric simple exclusion process six-vertex model integrable probability Macdonald symmetric functions Yang–Baxter equation


Barraquand, Guillaume; Borodin, Alexei; Corwin, Ivan; Wheeler, Michael. Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process. Duke Math. J. 167 (2018), no. 13, 2457--2529. doi:10.1215/00127094-2018-0019. https://projecteuclid.org/euclid.dmj/1535356817

Export citation


  • [1] A. Aggarwal, Convergence of the stochastic six-vertex model to the ASEP: Stochastic six-vertex model and ASEP, Math. Phys. Anal. Geom. 20 (2017), no. 3.
  • [2] A. Aggarwal, Current fluctuations of the stationary ASEP and six-vertex model, Duke Math. J. 167 (2018), 269–384.
  • [3] A. Aggarwal and A. Borodin, Phase transitions in the ASEP and stochastic six-vertex model, to appear in Ann. Probab., preprint, arXiv:1607.08684v1 [math.PR].
  • [4] G. Amir, I. Corwin, and J. Quastel, Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions, Comm. Pure Appl. Math. 64 (2011), 466–537.
  • [5] J. Baik, G. Barraquand, I. Corwin, and T. Suidan, Pfaffian Schur processes and last passage percolation in a half-quadrant, to appear in Ann. Probab., preprint, arXiv:1606.00525v3 [math.PR].
  • [6] J. Baik and E. M. Rains, Algebraic aspects of increasing subsequences, Duke Math. J. 109 (2001), 1–65.
  • [7] J. Baik and E. M. Rains, The asymptotics of monotone subsequences of involutions, Duke Math. J. 109 (2001), 205–281.
  • [8] G. Barraquand, A. Borodin, and I. Corwin, Half-space Macdonald processes, preprint, arXiv:1802.08210v1 [math.PR].
  • [9] R. J. Baxter, Exactly Solvable Models in Statistical Mechanics, Academic Press, London, 1982.
  • [10] L. Bertini and G. Giacomin, Stochastic Burgers and KPZ equations from particle systems, Comm. Math. Phys. 183 (1997), 571–607.
  • [11] D. Betea, M. Wheeler, and P. Zinn-Justin, Refined Cauchy/Littlewood identities and six-vertex model partition functions, II: Proofs and new conjectures, J. Algebraic Combin. 42 (2015), 555–603.
  • [12] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
  • [13] A. Borodin, On a family of symmetric rational functions, Adv. Math. 306 (2017), 973–1018.
  • [14] A. Borodin, Stochastic higher spin six vertex model and Macdonald measures, J. Math. Phys. 59 (2018), no. 023301.
  • [15] A. Borodin, A. Bufetov, and I. Corwin, Directed random polymers via nested contour integrals, Ann. Physics 368 (2016), 191–247.
  • [16] A. Borodin, A. Bufetov, and M. Wheeler, Between the stochastic six vertex model and Hall–Littlewood processes, to appear in J. Combin. Theory Ser. A, preprint, arXiv:1611.09486v1 [math.PR].
  • [17] A. Borodin and I. Corwin, Macdonald processes, Probab. Theory Related Fields 158 (2014), 225–400.
  • [18] A. Borodin, I. Corwin, and P. Ferrari, Free energy fluctuations for directed polymers in random media in $1+1$ dimension, Comm. Pure Appl. Math. 67 (2014), 1129–1214.
  • [19] A. Borodin, I. Corwin, and V. Gorin, Stochastic six-vertex model, Duke Math. J. 165 (2016), 563–624.
  • [20] A. Borodin, I. Corwin, L. Petrov, and T. Sasamoto, Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz, Comm. Math. Phys. 339 (2015), 1167–1245.
  • [21] A. Borodin and G. Olshanski, The ASEP and determinantal point processes, Comm. Math, Phys. 353 (2017), 853–903.
  • [22] A. Borodin and L. Petrov, Higher spin six vertex model and symmetric rational functions, Selecta Math. (N.S.) 24 (2018), 751–874.
  • [23] A. Borodin and E. M. Rains, Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys. 121 (2005), 291–317.
  • [24] P. Calabrese, P. Le Doussal, and A. Rosso, Free-energy distribution of the directed polymer at high temperature, Europhys. Lett. 90 (2010), no. 20002.
  • [25] S. Corteel and L. K. Williams, Tableaux combinatorics for the asymmetric exclusion process, Adv. in Appl. Math. 39 (2007), 293–310.
  • [26] S. Corteel and L. K. Williams, Staircase tableaux, the asymmetric exclusion process, and Askey–Wilson polynomials, Proc. Natl. Acad. Sci. USA 107 (2010), 6726–6730.
  • [27] S. Corteel and L. K. Williams, Tableaux combinatorics for the asymmetric exclusion process and Askey–Wilson polynomials, Duke Math. J. 159 (2011), 385–415.
  • [28] I. Corwin, The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), no. 1130001.
  • [29] I. Corwin and E. Dimitrov, Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles, Comm. Math. Phys., published electronically 4 May 2018.
  • [30] I. Corwin, J. Quastel, and D. Remenik, Continuum statistics of the $\mathrm{Airy}_{2}$ process, Comm. Math. Phys. 317 (2013), 347–362.
  • [31] I. Corwin and H. Shen, Open ASEP in the weakly asymmetric regime, Comm. Pure Appl. Math., published electronically 16 February 2018.
  • [32] B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26 (1993), 1493–1517.
  • [33] V. Dotsenko, 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, no. P07010.
  • [34] H. G. Duhart, P. Mörters, and J. Zimmer, The semi-infinite asymmetric exclusion process: Large deviations via matrix products, Potential Anal. 48 (2018), 301–323.
  • [35] S. Grosskinsky, Phase transitions in nonequilibrium stochastic particle systems with local conservation laws, Ph.D. dissertation, Technische Universität München, Munich, 2004.
  • [36] T. Gueudré and P. Le Doussal, Directed polymer near a hard wall and KPZ equation in the half-space, Europhys. Lett. 100 (2012), no. 26006.
  • [37] T. Halpin-Healy and K. A. Takeuchi, A KPZ cocktail—shaken, not stirred … toasting 30 years of kinetically roughened surfaces, J. Stat. Phys. 160 (2015), 794–814.
  • [38] M. Kardar, G. Parisi, and Y. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889–892.
  • [39] N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, and V. Terras, Algebraic Bethe ansatz approach to the asymptotic behavior of correlation functions, J. Stat. Mech. Theory Exp. 2009, no. P04003.
  • [40] G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Ann. of Math. (2) 156 (2002), 835–866.
  • [41] T. M. Liggett, Ergodic theorems for the asymmetric simple exclusion process, Trans. Amer. Math. Soc. 213 (1975), 237–261.
  • [42] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Clarendon, New York, 1995.
  • [43] N. O’Connell, T. Seppäläinen, and N. Zygouras, Geometric RSK correspondence, Whittaker functions and symmetrized random polymers, Invent. Math. 197 (2014), 361–416.
  • [44] A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random $3$-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), 581–603.
  • [45] F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.
  • [46] S. Parekh, The KPZ limit of ASEP with boundary, preprint, arXiv:1711.05297v1 [math.PR].
  • [47] E. M. Rains, Correlation functions for symmetrized increasing subsequences, preprint, arXiv:math/0006097v1 [math.CO].
  • [48] E. M. Rains, Multivariate quadratic transformations and the interpolation kernel, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), no. 019.
  • [49] T. Sasamoto and T. Imamura, Fluctuations of the one-dimensional polynuclear growth model in half-space, J. Stat. Phys. 115 (2004), 749–803.
  • [50] T. Sasamoto and H. Spohn, Exact height distributions for the KPZ equation with narrow wedge initial condition, Nuclear Phys. B 834 (2010), 523–542.
  • [51] T. Sasamoto and L. Williams, Combinatorics of the asymmetric exclusion process on a semi-infinite lattice, J. Comb. 5 (2014), 419–434.
  • [52] E. K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A. 21 (1988), 2375–2389.
  • [53] H. Spohn, Long range correlations for stochastic lattice gases in a nonequilibrium steady state, J. Phys. A 16 (1983), 4275–4291.
  • [54] H. Spohn, “The Kardar-Parisi-Zhang equation: a statistical physics perspective” in Stochastic Processes and Random Matrices, Oxford Univ. Press, Oxford, 2017, 177–227.
  • [55] C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151–174.
  • [56] C. A. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (1996), 727–754.
  • [57] C. A. Tracy and H. Widom, Matrix kernels for the Gaussian orthogonal and symplectic ensembles, Ann. Inst. Fourier (Grenoble) 55 (2005), 2197–2207.
  • [58] C. A. Tracy and H. Widom, Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), 129–154.
  • [59] C. A. Tracy and H. Widom, The asymmetric simple exclusion process with an open boundary, J. Math. Phys. 54 (2013), no. 103301.
  • [60] C. A. Tracy and H. Widom, The Bose gas and asymmetric simple exclusion process on the half-line, J. Stat. Phys. 150 (2013), 1–12.
  • [61] N. V. Tsilevich, Quantum inverse scattering method for the $q$-boson model and symmetric functions, Funct. Anal. Appl. 40 (2006), 207–217.
  • [62] M. Uchiyama, T. Sasamoto, and M. Wadati, Asymmetric simple exclusion process with open boundaries and Askey–Wilson polynomials, J. Phys. A 37 (2004), 4985–5002.
  • [63] M. Wheeler and P. Zinn-Justin, Refined Cauchy/Littlewood identities and six-vertex model partition functions, III: Deformed bosons, Adv. Math. 299 (2016), 543–600.