K. Mallick et S. Sandow, FINITE-DIMENSIONAL REPRESENTATIONS OF THE QUADRATIC ALGEBRA - APPLICATIONS TO THE EXCLUSION PROCESS, Journal of physics. A, mathematical and general, 30(13), 1997, pp. 4513-4526
We study the one-dimensional partially asymmetric simple exclusion pro
cess (ASEP) with open boundaries, that describes a system of hard-core
particles hopping stochastically on a chain coupled to reservoirs at
both ends. Derrida and coworkers showed in 1993 that the stationary pr
obability distribution of this model can be represented as a trace on
a quadratic algebra, closely related to the deformed oscillator-algebr
a. We construct all finite-dimensional irreducible representations of
this algebra. This enables us to compute the stationary bulk density a
s well as all correlation lengths for the ASEP on a set of special cur
ves of the phase diagram.