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.

arXiv comment62 pages, 6 figures

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Cited by in corpus (5)

• Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process
• 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
• Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model
• The q-Hahn PushTASEP