A reverse duality for the ASEP with open boundaries

TL;DR

This paper establishes a reverse duality between the open ASEP with non-conservative boundaries and a particle-dependent hopping process, linking measures of dual processes and clarifying stationary measure properties.

Contribution

It introduces a novel reverse duality framework for open ASEP, connecting shock measures with dual particle processes and enhancing understanding of stationary matrix product measures.

Findings

Duality relates measures, not expectations.

Shock measures evolve via dual particle processes.

Clarifies properties of stationary matrix product measures.

Abstract

We prove a duality between the asymmetric simple exclusion process (ASEP) with non-conservative open boundary conditions and an asymmetric exclusion process with particle-dependent hopping rates and conservative reflecting boundaries. This is a reverse duality in the sense that the duality function relates the measures of the dual processes rather than expectations. Specifically, for a certain parameter manifold of the boundary parameters of the open ASEP this duality expresses the time evolution of a family of shock product measures with $N$ microscopic shocks in terms of the time evolution of $N$ particles in the dual process. The reverse duality also elucidates some so far poorly understood properties of the stationary matrix product measures of the open ASEP given by finite-dimensional matrices.