Hidden Symmetries of Stochastic Models

Boyka Aneva

doi:10.3842/sigma.2007.068

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arXiv · 0705.2671

Hidden Symmetries of Stochastic Models

Boyka Aneva

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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.

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Boyka Aneva. 2007-05-18. Hidden Symmetries of Stochastic Models. https://doi.org/10.3842/sigma.2007.068

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