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” $$\alpha \beta =q^N\gamma \delta ,$$ 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, $$\varphi _1,$$ is defined on the entire algebra, and determines stationary probabilities for large systems on $$L\ge N+1$$ sites. The other functional, $$\varphi _0,$$ is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on $$L< N+1$$ sites. Functional $$\varphi _0$$ vanishes on non-constant Askey–Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey–Wilson polynomials.