Between the stochastic six vertex model and Hall-Littlewood processes

TL;DR

This paper establishes a deep connection between the stochastic six vertex model's height function distribution and Hall-Littlewood processes, revealing new insights into their joint distributions and limiting behaviors.

Contribution

It demonstrates the equivalence of height function distributions in the stochastic six vertex model and Hall-Littlewood processes, including in a continuous limit with RSK-type evolution.

Findings

Joint distribution of height functions matches Hall-Littlewood process lengths

Distribution equivalence persists in the continuous limit

Height function field shares distribution with evolving Hall-Littlewood partitions

Abstract

We prove that the joint distribution of the values of the height function for the stochastic six vertex model in a quadrant along a down-right path coincides with that for the lengths of the first columns of partitions distributed according to certain Hall-Littlewood processes. In the limit when one of the quadrant axes becomes continuous, we also show that the two-dimensional random field of the height function values has the same distribution as the lengths of the first columns of partitions from certain ascending Hall-Littlewood processes evolving under a Robinson-Schensted-Knuth type Markovian evolution.