The $q$-TASEP with a random initial condition

A standard approach for studying fluctuations of one-dimensional Kardar–Parisi–Zhang models, which include the ASEP and the $q$-TASEP, is to write a formula for the $q$-deformed moments and construct their generating function. This approach works well for an initial condition of the step type but not for a random initial condition (including the stationary case): in this case only the first few moments are finite and the rest diverge. We previously presented a method for overcoming this difficulty using the Ramanujan summation formula and the Cauchy determinant for the theta functions. Here, we present an alternative approach for the $q$-TASEP without using these relations.