Hidden Symmetries of Stochastic Models

Boyka Aneva

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

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.

Détail

Keywords: stohastic models; tridiagonal algebra; Askey–Wilson polynomials.

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