Torpid Mixing of Markov Chains for the Six-vertex Model on $\mathbb{Z}^2$

TL;DR

This paper proves that certain Markov chain algorithms for the six-vertex model on finite square lattices mix slowly in specific parameter regimes, highlighting limitations in their efficiency for these phases.

Contribution

It is the first to establish torpid mixing of Glauber dynamics and the directed-loop algorithm for the six-vertex model in ferroelectric and anti-ferroelectric phases.

Findings

Markov chains are torpidly mixing in ferroelectric phase

Markov chains are torpidly mixing in anti-ferroelectric phase

Results apply to finite regions of the square lattice

Abstract

In this paper, we study the mixing time of two widely used Markov chain algorithms for the six-vertex model, Glauber dynamics and the directed-loop algorithm, on the square lattice $\mathbb{Z}^2$. We prove, for the first time that, on finite regions of the square lattice these Markov chains are torpidly mixing under parameter settings in the ferroelectric phase and the anti-ferroelectric phase.