Matrix product construction for Koornwinder polynomials and fluctuations of the current in the open ASEP

TL;DR

This paper develops a matrix product approach to connect Koornwinder polynomials with current fluctuations in the open ASEP, revealing new algebraic structures and conjecturing a link to cumulant generating functions.

Contribution

It introduces a novel matrix product construction for symmetric Koornwinder polynomials related to the open ASEP's current fluctuations.

Findings

Koornwinder polynomials emerge as normalization of deformed ground states

Matrix product construction for these polynomials is established

Numerical evidence suggests a relation between cumulant generating functions and Koornwinder polynomials

Abstract

Starting from the deformed current-counting transition matrix for the open boundary ASEP, we prove that with a further deformation, the symmetric Koornwinder polynomials for partitions with equal row lengths appear as the normalisation of the twice deformed ground state. We give a matrix product construction for this ground state and the corresponding symmetric Koornwinder polynomials. Based on the form of this construction and numerical evidence, we conjecture a relation between the generating function of the cumulants of the current, and a certain limit of the symmetric Koornwinder polynomials.