Bi-orthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process

Richard Brak; William Moore

doi:10.1088/1751-8113/48/31/315205

paper

Bi-orthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process

arXiv:1412.7235 · doi:10.1088/1751-8113/48/31/315205

Abstract

We reformulate the Corteel-Williams equations for the stationary state of the two parameter Asymmetric Simple Exclusion Process (TASEP) as a linear map $L (\cdot)$ , 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 $L (P_{n} (e_{1}) \otimes Q_{m} (e_{2})) = Λ_{n} δ_{n, m}$ . 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 $L (P_{n} e_{1} Q_{m})$ and $L (P_{n} e_{2} Q_{m})$ 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 $L (P_{n} e_{1} e_{2} Q_{m})$ defines a tridiagonal matrix which makes the connection with Chebyshev-like orthogonal polynomials.

Bi-orthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process

Abstract

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