Stochastic models on a ring and quadratic algebras. The three-species diffusion problem

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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. We consider only exclusion processes and restrict the type of algebras such that one has recurrence relations for traces of words of different lengths. This is possible only if the rates satisfy certain compatibility conditions. These conditions are derived and explicit representations of the generators of the quadratic algebras are given.

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