Finite Dimensional Representations of the Quadratic Algebra: Applications to the Exclusion Process

TL;DR

This paper constructs all finite dimensional irreducible representations of a quadratic algebra related to the ASEP model, enabling exact calculations of stationary densities and correlation lengths on specific phase diagram curves.

Contribution

It provides a complete classification of finite dimensional representations of the algebra, facilitating exact analysis of the ASEP's stationary properties.

Findings

Computed stationary bulk density for specific phase diagram curves

Determined correlation lengths explicitly

Connected algebraic representations to physical observables

Abstract

We study the one dimensional partially asymmetric simple exclusion process (ASEP) with open boundaries, that describes a system of hard-core particles hopping stochastically on a chain coupled to reservoirs at both ends. Derrida, Evans, Hakim and Pasquier [J. Phys. A 26, 1493 (1993)] have shown that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra, closely related to the deformed oscillator-algebra. We construct all finite dimensional irreducible representations of this algebra. This enables us to compute the stationary bulk density as well as all correlation lengths for the ASEP on a set of special curves of the phase diagram.