Stochastic Models on a Ring and Quadratic Algebras. The Three Species Diffusion Problem

TL;DR

This paper investigates the stationary states of a stochastic process on a ring using quadratic algebra representations, deriving conditions for exclusion processes and solving recurrence relations to understand the algebraic structure.

Contribution

It introduces a method to express stationary states via quadratic algebra traces and derives compatibility conditions for exclusion process rates, providing algebraic solutions.

Findings

Derived compatibility conditions for exclusion process rates

Solved recurrence relations for algebra representations

Expressed stationary states using traces of algebra monomials

Abstract

The stationary state of a stochastic process on a ring can be expressed using traces of monomials of an associative algebra defined by quadratic relations. If one considers only exclusion processes one can restrict the type of algebras and obtain recurrence relations for the traces. This is possible only if the rates satisfy certain compatibility conditions. These conditions are derived and the recurrence relations solved giving representations of the algebras.