N-species stochastic models with boundaries and quadratic algebras

TL;DR

This paper develops a matrix product approach to find stationary distributions of N-species stochastic models on chains with various boundary conditions, revealing differences from traditional Bethe Ansatz methods especially for N>2.

Contribution

It introduces a new algebraic framework for N-species models with boundaries, extending matrix product techniques beyond the well-studied two-species case.

Findings

Derived stationary distributions using quadratic algebras for N-species models.

Identified differences between N=2 and N>2 cases in algebraic structures.

Connected stochastic models to quantum chain ground states.

Abstract

Stationary probability distributions for stochastic processes on linear chains with closed or open ends are obtained using the matrix product Ansatz. The matrices are representations of some quadratic algebras. The algebras and the types of representations considered depend on the boundary conditions. In the language of quantum chains we obtain the ground state of N-state quantum chains with free boundary conditions or with non-diagonal boundary terms at one or both ends. In contrast to problems involving the Bethe Ansatz, we do not have a general framework for arbitrary N which when specialized, gives the known results for N=2; in fact, the N=2 and N>2 cases appear to be very different.