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

Abstract

We analyze the mixing time of Glauber dynamics for the six-vertex model in the ferroelectric and antiferroelectric phases. Speci cally, 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 rst rigorous result about the mixing time of Glauber dynamics for the six-vertex model in the ferroelectric phase. Furthermore, our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models. We also analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and signi cantly 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.