 
Summary: J. Phys. A: Math. Gen. 31 (1998) 833843. Printed in the UK PII: S03054470(98)844739
Stochastic models on a ring and quadratic algebras. The
threespecies diffusion problem
Peter F Arndt, Thomas Heinzel and Vladimir Rittenberg
SISSA, Via Beirut 24, 34014 Trieste, Italy
Received 22 May 1997, in final form 15 July 1997
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.
1. Introduction
In a previous paper [1] we considered the application of quadratic algebras to stochastic
problems with closed or open boundaries. Here we study the case of periodic boundary
conditions. We are again interested in the (unnormalized) probability distributions describing
stationary states. In the language of quantum chains, we seek ground states which have zero
momentum and energy. Much work has already been done in seeking matrixproduct states
in the case of periodic chains [24]. In the language of [1] in these papers polynomial
algebras which have a trace operation were used. (Polynomial algebras are quadratic
