Stationary q-TASEP and elliptic determinant

The 1D totally asymmetric simple exclusion process (TASEP) is a stochastic interacting particle system on Z, in which particles perform Poisson random walks in one direction under volume exclusion. It shows interesting scaling behaviors related to the Kardar-Parisi-Zhang universality and has many applications in physics, biology, traffic flow and so on. The TASEP is exactly solvable. For example, the current fluctuation of the TASEP can be studied by its connection to random matrix theory. 

The $q-TASEP$ is a generalization of TASEP in which the hopping rate of each particle is generalized to $1-q^gap$, where $0

In this presentation, we present a Fredholm determinant formula for the random initial condition of $q-TASEP$. We first explain that the $q-TASEP$ can be described by a two-sided version of the $q-Whittaker$ process. Then we show that the $q-Laplace$ transform of a particle position can be calculated by using the Ramanujan’s bilateral summation formula and the Cauchy determinant formula for the theta function, resulting in its Fredholm determinant expression. Finally we show that the fluctuations are described by the universal stationary $KPZ$ distribution (Baik-Rains distribution) in the long time limit. 

Reference
T. Imamura, T. Sasamoto, Fluctuations for stationary $q-TASEP$, arXiv: 1701.05991.