Blocks in the Asymmetric Simple Exclusion Process

TL;DR

This paper extends formulas for particle position probabilities in the asymmetric simple exclusion process to cases involving blocks of particles, generalizing previous results and combinatorial identities for broader initial conditions.

Contribution

It introduces generalized combinatorial identities and formulas for block configurations in the ASEP, expanding the analytical framework beyond single particles.

Findings

Formulas for block particle positions derived

Generalized combinatorial identities established

Extended integral representations for block configurations

Abstract

In earlier work the authors obtained formulas for the probability in the asymmetric simple exclusion process that the $m$th particle from the left is at site $x$ at time $t$. They were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In the present work these results are generalized to the case where the $m$th particle is the left-most one in a contiguous block of $L$ particles. The earlier work depended in a crucial way on two combinatorial identities, and the present work begins with a generalization of these identities to general $L$.