On the $q$-TASEP with a random initial condition

TL;DR

This paper develops an alternative method to analyze the $q$-TASEP with random initial conditions, overcoming divergence issues in moments by directly handling the $q$-deformed Laplace transform, without relying on Ramanujan's summation or theta functions.

Contribution

It introduces a new approach for $q$-TASEP with random initial conditions that avoids the divergence problems of moments, differing from previous methods.

Findings

Provides a new analytical framework for $q$-TASEP with random initial conditions.

Circumvents divergence issues in moments for random initial conditions.

Offers an alternative to previous methods relying on Ramanujan's summation and theta functions.

Abstract

When studying fluctuations of models in the 1D KPZ class including the ASEP and the $q$-TASEP, a standard approach has been to first write down a formula for $q$-deformed moments and constitute their generating function. This works well for the step initial condition, but there is a difficulty for a random initial condition (including the stationary case): in this case only the first few moments are finite and the rest diverge. In a previous work [16], we presented a method dealing directly with the $q$-deformed Laplace transform of an observable, in which the above difficulty does not appear. There the Ramanujan's summation formula and the Cauchy determinant for the theta functions play an important role. In this note, we give an alternative approach for the $q$-TASEP without using them.