Bi-orthogonal Polynomials and the Five parameter Asymmetric Simple Exclusion Process

arXiv1902.06373MaRDI QIDQ6314187

Wynton E. Moore, Rolv Bræk

Publication date: 17 February 2019

Abstract: We apply the bi-moment determinant method to compute a representation of the matrix product algebra -- a quadratic algebra satisfied by the operators $mathbfd$ and $mathbfe$ -- for the five parameter ($alpha$, $$, $gamma$, $delta$ and $q$) Asymmetric Simple Exclusion Process. This method requires an $LDU$ decomposition of the ``bi-moment matrix. The decomposition defines a new pair of basis vectors sets, the `boundary basis'. This basis is defined by the action of polynomials $P_n$ and $Q_n$ on the quantum oscillator basis (and its dual). Theses polynomials are orthogonal to themselves (ie. each satisfy a three term recurrence relation) and are orthogonal to each other (with respect to the same linear functional defining the stationary state). Hence termed `bi-orthogonal'. With respect to the boundary basis the bi-moment matrix is diagonal and the representation of the operator $mathbfd+mathbfe$ is tri-diagonal. This tri-diagonal matrix defines another set of orthogonal polynomials very closely related to the the Askey-Wilson polynomials (they have the same moments).

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