# On matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the singular case

## TL;DR

This paper introduces a new approach using linear functionals to analyze the stationary states of the Asymmetric Simple Exclusion Process in the singular boundary case, where traditional matrix product methods fail.

## Contribution

It develops a novel functional-based framework that extends the matrix product ansatz to singular cases of ASEP with open boundaries, connecting to Askey-Wilson polynomials.

## Key findings

- Functional $_1$ determines stationary probabilities for large systems.
- Functional $_0$ applies to small systems and relates to Askey-Wilson polynomials.
- The approach generalizes the matrix product ansatz to singular boundary conditions.

## 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, Evans, Hakim and Pasquier 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 $L\geq N+1$ 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.