The PASEP at q=-1

TL;DR

This paper analyzes the PASEP model at q=-1 using a two-dimensional algebraic representation, revealing that key physical quantities behave reasonably despite the unphysical parameter value, and compares this to known representations.

Contribution

It introduces a two-dimensional representation for PASEP at q=-1, enabling straightforward calculations and comparison with existing models, and discusses extensions and limitations of different representations.

Findings

Normalization, current, and density are easily computed at q=-1.

Quantities behave reasonably despite unphysical parameter values.

Comparison with q between 0 and 1 shows similarities and differences.

Abstract

We investigate the partially asymmetric exclusion process (PASEP) with open boundaries when the reverse hopping rate of particles q=-1, using a representation of the PASEP algebra related to the al-Salam Chihara polynomials. When q=-1 the representation is two-dimensional, which allows for straightforward calculation of the normalization, current and density. We note that these quantities behave in an a priori reasonable manner in spite of the apparently unphysical value of q as the input, alpha, and output, beta, rates are varied over the physical range of 0 to 1. As is well known, another two dimensional representation exists when 0<q<1 and abq=1, where a=(1-q)/beta -1 and b=(1-q)/beta-1, and we compare the behaviour at q=-1 with this. An extension to generalized boundary conditions where particles may enter and exit at both ends is briefly outlined. We also note that a different representation related to the q-harmonic oscillator does not admit a straightforward truncation when q=-1 and discuss why this is the case from the perspective of a lattice path interpretation of the PASEP normalization.