Bi-orthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process

We reformulate the Corteel-Williams equations for the stationary state of the two parameter Asymmetric Simple Exclusion Process (TASEP) as a linear map $\mathcal{L}(\,\cdot\,)$, acting on a tensor algebra built from a rank two free module with basis $\{e_1,e_2\}$. From this formulation we construct a pair of sequences, $\{P_n(e_1)\}$ and $\{Q_m(e_2)\}$, of bi-orthogonal polynomials (BiOPS), that is, they satisfy $\mathcal{L}(P_n(e_1)\otimes Q_m(e_2))=\Lambda_n\delta_{n,m}$. The existence of the sequences arises from the determinant of a Pascal triangle like matrix of polynomials. The polynomials satisfy first order (uncoupled) recurrence relations. We show that the two first moments $\mathcal{L}(P_n\, e_1\, Q_m)$ and $\mathcal{L}(P_n\, e_2\, Q_m)$ give rise to a matrix representation of the ASEP diffusion algebra and hence provide an understanding of the origin of the matrix product Ansatz. The second moment $\mathcal{L}(P_n\, e_1 e_2\,Q_m )$ defines a tridiagonal matrix which makes the connection with Chebyshev-like orthogonal polynomials.