On Matrix Product Ansatz for Asymmetric Simple Exclusion Process with Open Boundary in the Singular Case

Abstract

We study a substitute for the matrix product ansatz for asymmetric simple exclusion process with open boundary in the “singular case” $$\\alpha \\beta =q^N\\gamma \\delta ,$$\n when the standard form of the matrix product ansatz of Derrida et al. (J Phys A 26(7):1493–1517, 1993) does not apply. In our approach, the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra. One of the functionals, $$\\varphi _1,$$\n is defined on the entire algebra, and determines stationary probabilities for large systems on $$L\\ge N+1$$\n sites. The other functional, $$\\varphi _0,$$\n is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on $$L< N+1$$\n sites. Functional $$\\varphi _0$$\n vanishes on non-constant Askey–Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey–Wilson polynomials.