Strong law of large numbers for the stochastic six vertex model

TL;DR

This paper proves a strong law of large numbers for the inhomogeneous stochastic six vertex model with periodic initial data, showing almost sure convergence to a deterministic shape by mapping to a deformed Hammersley process and using ergodic theory.

Contribution

It introduces a new approach using a colored model and Boolean-type product to establish almost sure convergence in the stochastic six vertex model.

Findings

Almost sure convergence to a deterministic limit shape.

Mapping to a deformed Hammersley process facilitates analysis.

A new colored model construction simplifies previous methods.

Abstract

We consider the inhomogeneous stochastic six vertex model with periodicity starting from step initial data. We prove that it converges almost surely to a deterministic limit shape. For the proof, we map the stochastic six vertex model to a deformed version of the discrete Hammersley process. Then we construct a colored version of the model and apply Liggett's superadditive ergodic theorem. The construction of the colored model includes a new idea using a Boolean-type product, which generalizes and simplifies the method used in arXiv:2204.11158.