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

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1535356817

Digital Object Identifier
doi:10.1215/00127094-2018-0019

Mathematical Reviews number (MathSciNet)
MR3855355

Zentralblatt MATH identifier
06970973

Subjects
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)

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

Citation

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


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