Abstract
We consider the $q$-Hahn TASEP which is a three-parameter family of discrete time interacting particle systems. The particles jump to the right independently according to a certain $q$-Binomial distribution with parallel updates. It is a generalization of the totally asymmetric simple exclusion process (TASEP) on $\mathbb{Z}$. For step initial condition, we prove that the current fluctuation of $q$-Hahn TASEP at time $\tau$ is of order $\tau^{1/3}$ and asymptotically distributed as the GUE Tracy–Widom distribution. We verify the KPZ scaling theory conjecture for the $q$-Hahn TASEP.
Highlights
In the totally asymmetric simple exclusion process (TASEP) on the one-dimensional integer lattice Z, particles with vacant right neighbour jump to the right by 1 according to independent Poisson processes with unit rate
A Fredholm determinant formula was given in [6] for the q-Laplace transform of the particle position. q-TASEP belongs to the Kardar–Parisi–Zhang (KPZ) universality class
Based on the formula of [6], Ferrari and Vetoshowed in [16] that the large time current fluctuations are governed by the (GUE) Tracy–Widom distribution
Summary
In the totally asymmetric simple exclusion process (TASEP) on the one-dimensional integer lattice Z, particles with vacant right neighbour jump to the right by 1 according to independent Poisson processes with unit rate. Using the duality of the q-Hahn Boson process and the q-Hahn TASEP, Corwin derived a Fredholm determinant formula for the q-Laplace transform of the particle position in q-Hahn TASEP with step initial condition in [11], the proof was recently simplified by Barraquand in [1]. This formula is used as a starting point of the asymptotic analysis carried out in the present paper, see Theorem 4.1 below.
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