# Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases

## TL;DR

This paper proves that Glauber dynamics for the six-vertex model in ordered phases can require exponential time to mix, revealing fundamental limitations of local Markov chains in these regimes.

## Contribution

It provides the first rigorous bounds on the slow mixing of Glauber dynamics in the ferroelectric phase of the six-vertex model, extending understanding in ordered phases.

## Key findings

- Glauber dynamics mixes exponentially slow in the ferroelectric phase.
- Boundary conditions can induce slow mixing in the ordered phases.
- New techniques relate correlated random walks to lattice path models.

## Abstract

The six-vertex model in statistical physics is a weighted generalization of the ice model on $\mathbb{Z}^2$ (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model depicts its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the first rigorous result bounding the mixing time of Glauber dynamics in the ferroelectric phase. Our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models (or routings). We analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and significantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks.