Determinant solution for the Totally Asymmetric Exclusion Process with parallel update II. Ring geometry

TL;DR

This paper derives a determinant formula for the time-dependent transition probabilities in the TASEP with parallel update on a ring, using Bethe ansatz and complex analysis techniques, linking to existing results and providing new generating functions.

Contribution

It introduces a novel method to sum over Bethe roots and constructs the identity operator, enabling calculation of evolution operator matrix elements for TASEP on a ring.

Findings

Determinant expression for transition probabilities derived

Connection established with previous ring geometry results

Generated functions for particle configurations and total distance obtained

Abstract

Using the Bethe ansatz we obtain the determinant expression for the time dependent transition probabilities in the totally asymmetric exclusion process with parallel update on a ring. Developing a method of summation over the roots of Bethe equations based on the multidimensional analogue of the Cauchy residue theorem, we construct the resolution of the identity operator, which allows us to calculate the matrix elements of the evolution operator and its powers. Representation of results in the form of an infinite series elucidates connection to other results obtained for the ring geometry. As a byproduct we also obtain the generating function of the joint probability distribution of particle configurations and the total distance traveled by the particles.