Colored Line Ensembles for Stochastic Vertex Models

TL;DR

This paper constructs coupled line ensembles for colored stochastic vertex models, revealing their law and Gibbs properties, and connects these models to well-known integrable systems like the log-gamma polymer and colored ASEP.

Contribution

It introduces a family of coupled line ensembles for $U_q( ext{sl}_{n+1})$ models with explicit Gibbs properties and links to classical integrable models.

Findings

The joint law of top curves matches colored height functions.

Line ensembles satisfy an explicit Gibbs property.

Degeneration to log-gamma polymer and convergence to colored ASEP.

Abstract

In this paper we assign a family of $n$ coupled line ensembles to any $U_q (\widehat{\mathfrak{sl}}_{n+1})$ colored stochastic fused vertex model, which satisfies two properties. First, the joint law of their top curves coincides with that of the colored height functions for the vertex model. Second, the $n$ line ensembles satisfy an explicit Gibbs property prescribing their laws if all but a few of their curves are conditioned upon. We further describe several examples of such famlies of line ensembles, including the ones for the colored stochastic six-vertex and $q$-boson models. The appendices (which may be of independent interest) include an explanation of how the $U_q (\widehat{\mathfrak{sl}}_{n+1})$ colored stochastic fused vertex model degenerates to the log-gamma polymer, and an effective rate of convergence of the colored stochastic six-vertex model to the colored ASEP.