Continued fractions and the partially asymmetric exclusion process

We note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process (PASEP) lends itself to interpretation as the transfer matrix for weighted Motzkin lattice paths. A continued-fraction ('J fraction') representation of the lattice-path-generating function is particularly well suited to discussing the PASEP, for which the paths have height-dependent weights. We show that this not only allows a succinct derivation of the normalization and correlation lengths of the PASEP, but also reveals how finite-dimensional representations of the PASEP algebra, valid only along special lines in the phase diagram, relate to the general solution that requires an infinite-dimensional representation.