Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases
The six-vertex model in statistical physics is a weighted generalization of the ice model on 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 represents 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.
Correlated random walk
Markov chain Monte Carlo
Six-vertex model
Theory of computation~Random walks and Markov chains
37:1-37:20
RANDOM
A full version of the paper is available at https://arxiv.org/abs/1904.01495.
Matthew
Fahrbach
Matthew Fahrbach
School of Computer Science, Georgia Institute of Technology, Atlanta, Georgia, USA
Supported in part by an NSF Graduate Research Fellowship under grant DGE-1650044.
Dana
Randall
Dana Randall
School of Computer Science, Georgia Institute of Technology, Atlanta, Georgia, USA
Supported in part by NSF grants CCF-1637031 and CCF-1733812.
10.4230/LIPIcs.APPROX-RANDOM.2019.37
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Matthew Fahrbach and Dana Randall
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