Fluctuations for stationary $q$-TASEP

Understand

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.