Hidden Symmetries of Stochastic Models

TL;DR

This paper explores the algebraic symmetries in stochastic models, particularly the role of quantum groups and Askey-Wilson algebra, to achieve exact solutions for boundary-driven diffusion processes.

Contribution

It identifies the Askey-Wilson algebra as a key symmetry in boundary processes of stochastic models, enabling exact solutions.

Findings

The quadratic algebra defines a noncommutative space with $SU_q(n)$ symmetry.

Boundary operators generate a tridiagonal algebra related to Askey-Wilson polynomials.

The Askey-Wilson algebra allows exact solutions of the boundary problem.

Abstract

In the matrix product states approach to $n$ species diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process. The quadratic algebra defines a noncommutative space with a $SU_q(n)$ quantum group action as its symmetry. Boundary processes amount to the appearance of parameter dependent linear terms in the algebraic relations and lead to a reduction of the $SU_q(n)$ symmetry. We argue that the boundary operators of the asymmetric simple exclusion process generate a tridiagonal algebra whose irriducible representations are expressed in terms of the Askey-Wilson polynomials. The Askey-Wilson algebra arises as a symmetry of the boundary problem and allows to solve the model exactly.