Stationary Higher Spin Six Vertex Model and $q$-Whittaker measure

TL;DR

This paper analyzes the stationary Higher Spin Six Vertex Model on a lattice, deriving exact formulas for the height distribution, and establishes asymptotic fluctuation results, including Baik-Rains fluctuations, with applications to related models.

Contribution

It introduces a family of translation invariant measures and provides exact formulas for the height distribution, advancing understanding of the model's asymptotic behavior.

Findings

Recovered Baik-Rains fluctuations with exponent 1/3 along the characteristic line.

Derived exact formulas for the one-point height distribution.

Extended analysis to degenerations like the $q$-Hahn process and Exponential Jump Model.

Abstract

In this paper we consider the Higher Spin Six Vertex Model on the lattice $\mathbb{Z}_{\geq 2} \times \mathbb{Z}_{\geq 1}$. We first identify a family of translation invariant measures and subsequently we study the one point distribution of the height function for the model with certain random boundary conditions. Exact formulas we obtain prove to be useful in order to establish the asymptotic of the height distribution in the long space-time limit for the stationary Higher Spin Six Vertex Model. In particular, along the characteristic line we recover Baik-Rains fluctuations with size of characteristic exponent $1/3$. We also consider some of the main degenerations of the Higher Spin Six Vertex Model and we adapt our analysis to the relevant cases of the $q$-Hahn particle process and of the Exponential Jump Model.