Fluctuations for stationary $q$-TASEP

Abstract

We consider the $q$-totally asymmetric simple exclusion process ($q$-TASEP) in the stationary regime and study the fluctuation of the position of a particle. We first observe that the problem can be studied as a limiting case of an $N$-particle $q$-TASEP with a random initial condition and with particle dependent hopping rate. Then we explain how this $N$-particle $q$-TASEP can be encoded in a dynamics on a two-sided Gelfand-Tsetlin cone described by a two-sided $q$-Whittaker process and present a Fredholm determinant formula for the $q$-Laplace transform of the position of a particle. Two main ingredients in its derivation is the Ramanujan's bilateral summation formula and the Cauchy determinant identity for the theta function with an extra parameter. Based on this we establish that the position of a particle obeys the universal stationary KPZ distribution (the Baik-Rains distribution) in the long time limit.

62 pages, 6 figures

References in corpus (8)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Asymptotics in ASEP with Step Initial Condition
• Exact scaling functions for one-dimensional stationary KPZ growth
• Fluctuation properties of the TASEP with periodic initial configuration
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Bethe ansatz solution of zero-range process with nonuniform stationary state
• A phase transition for $q$-TASEP with a few slower particles
• Stochastic higher spin six vertex model and Madconald measures

Cited by in corpus (2)

• Stationary Higher Spin Six Vertex Model and $q$-Whittaker measure
• Distributions of a particle's position and their asymptotics in the $q$-deformed totally asymmetric zero range process with site dependent jumping rates