Limit Shapes of the Stochastic Six Vertex Model

TL;DR

This paper studies the limit shapes of the stochastic 6-vertex model on a cylinder, showing they satisfy a Burger-type equation, and connects these solutions to known results in the infinite circumference limit.

Contribution

It derives the limit shape equations for the stochastic 6-vertex model on a cylinder and links finite and infinite cases, extending prior work.

Findings

Limit shapes satisfy Burger-type equations.

Solutions for step initial conditions are characterized.

Connection to known infinite-cylinder solutions when circumference grows.

Abstract

We show that limit shapes for the stochastic 6-vertex model on a cylinder with the uniform boundary state on one end are solutions to the Burger type equation. Solutions to these equations are studied for step initial conditions. When the circumference of the cylinder goes to infinity the solution corresponding to critical initial densities coincides with the one found by Borodin, Corwin and Gorin.