Biorthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process

Abstract

The diffusion algebra equations of the stationary state of the three parameter Asymmetric Simple Exclusion Process are represented as a linear functional, acting on a tensor algebra. From the linear functional, a pair of sequences (P and Q) of monic polynomials are constructed which are bi-orthogonal, that is, they are orthogonal with respect to each other and not necessarily themselves. The uniqueness and existence of the pair of sequences arises from the determinant of the bi-moment matrix whose elements satisfy a pair of q-recurrence relations. The determinant is evaluated using an LDU-decomposition. If the action of the linear functional is represented as an inner product, then the action of the polynomials Q on a boundary vector V, generates a basis whose orthogonal dual vectors are given by the action of P on the dual boundary vector W}. This basis gives the representation of the algebra which is associated with the Al-Salam-Chihara polynomials obtained by Sasamoto.

Several theorems associated with the three parameter asymmetric simple exclusion process are proven combinatorially. The theorems involve the linear functional which, for the three parameter case, is a substitution morphism on a q-Weyl algebra. The two polynomial sequences, P and Q, are represented in terms of q-binomial lattice paths. A combinatorial representation for the value of the linear functional defining the matrix elements of a bi-moment matrix is established in terms of the value of a q-rook polynomial and utilised to provide combinatorial proofs for results pertaining to the linear functional. Combinatorial proofs are provided for theorems in terms of the p,q-binomial coefficients, which are closely related to the combinatorics of the three parameter ASEP.

The results for the three parameter diffusion algebra of the Asymmetric Simple Exclusion Process are extended to five parameters. A pair of basis changes are derived from the LDU decomposition of the bi-moment matrix. In order to derive the LDU decomposition a recurrence relation satisfied by the lower triangular matrix elements is conjectured. Associated with this pair of bases are three sequences of orthogonal polynomials. The first pair of orthogonal polynomials generate the new basis vectors (the boundary basis) by their action on the boundary vectors (written is the standard basis), whilst the third orthogonal polynomials are essentially the Askey-Wilson polynomials. All theses results are ultimately related to the LDU decomposition of a matrix.