Free boundary q-Whittaker and Hall-Littlewood processes

TL;DR

This paper investigates free boundary $q$-Whittaker and Hall--Littlewood processes, establishing symmetry properties, deriving contour integral formulas, and connecting these processes to vertex models and Koornwinder polynomials.

Contribution

It introduces a new symmetry result for the free boundary $q$-Whittaker process, provides contour integral formulas, and links Hall--Littlewood processes to vertex models and Koornwinder polynomials.

Findings

Proved a $(q,t)$ symmetry after a random shift for a key observable.

Derived contour integral formulas for the free boundary $q$-Whittaker process.

Established a connection between Hall--Littlewood processes and a quasi-open six vertex model.

Abstract

We study the free boundary $q$-Whittaker and Hall--Littlewood processes, two probability measures on sequences of partitions. We prove that a certain observable of the free boundary $q$-Whittaker process exhibits a $(q,t)$ symmetry after a random shift, generalizing a previous result of Imamura, Mucciconi, and Sasamoto, and an extension of that result due to the first author. Our proof is completely different, and as part of our proof, we find contour integral formulas for the free boundary $q$-Whittaker process. We also show a matching between certain observables in the free boundary Hall--Littlewood process and a quasi-open six vertex model, and explain how work of Finn and Vanicat gives an evaluation of a bounded sum over skew Hall--Littlewood functions as a rectangular Koornwinder polynomial.