Continued Fractions and the Partially Asymmetric Exclusion Process

TL;DR

This paper explores the connection between continued fractions, Motzkin paths, and the algebraic structure of the PASEP, providing new insights into its normalization, correlation lengths, and finite versus infinite-dimensional representations.

Contribution

It introduces a continued fraction approach to analyze the PASEP, linking lattice path weights to algebraic representations and phase diagram features.

Findings

Derived a succinct expression for PASEP normalization using continued fractions.

Connected finite-dimensional representations to the general infinite-dimensional solution.

Provided a new interpretation of the PASEP algebra via Motzkin path transfer matrices.

Abstract

We note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process (PASEP) lends itself to interpretation as the transfer matrix for weighted Motzkin lattice paths. A continued fraction ("J-Fraction") representation of the lattice path generating function is particularly well suited to discussing the PASEP, for which the paths have height dependent weights. We show that this not only allows a succinct derivation of the normalisation and correlation lengths of the PASEP, but also reveals how finite-dimensional representations of the PASEP algebra, valid only along special lines in the phase diagram, relate to the general solution that requires an infinite-dimensional representation.