On dualities for SSEP and ASEP with open boundary conditions

TL;DR

This paper explores duality relations in simple exclusion processes with open boundaries, introducing a unified approach using spin and bosonic operators, and reveals new boundary conditions for SSEP and complex dualities for ASEP.

Contribution

It presents a unified framework for duality relations in SSEP and ASEP with open boundaries, including new boundary conditions and the use of spin and bosonic operators.

Findings

For SSEP, additional copying sites are needed at boundaries.

Unified approach using spin and bosonic operators for duality.

Complex dual processes are identified for ASEP with open boundaries.

Abstract

Duality relations for simple exclusion processes with general open boundaries are discussed. It is shown that a combination of spin operators and bosonic operators enables us to have an unified discussion for the duality relations with the open boundaries. As for the symmetric simple exclusion process (SSEP), more general results than previous studies are obtained; it is clarified that not only the absorbing sites, but also additional sites, called copying sites, are needed for the boundaries in the dual process for the SSEP. The role of the copying sites is to conserve information about the particle states on the boundary sites. The similar discussions are applied to the asymmetric simple exclusion process (ASEP), in which the q-analogues are employed, and it is clarified that the ASEP with open boundaries has a complicated dual process on the boundaries.