The Semi-Infinite Asymmetric Exclusion Process: Large Deviations via Matrix Products

TL;DR

This paper derives a large deviation principle for the particle density in a semi-infinite asymmetric exclusion process using matrix product representations, introducing a novel spectral and combinatorial approach.

Contribution

It introduces a rigorous method combining spectral and combinatorial techniques to analyze large deviations in models described by matrix products.

Findings

Explicit large deviation rate function for particle density

Applicable spectral and combinatorial methodology

Enhanced understanding of phase transition effects

Abstract

We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. Liggett (1975) has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source is below a critical value, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of Derrida et al. (1993) it was shown by Gro{\ss}kinsky (2004) that these correlations can be described by means of a matrix product representation. In this paper we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and has the potential to be applicable to other models described by matrix products.