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

Authors Matthew Fahrbach, Dana Randall

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Author Details

Matthew Fahrbach
  • School of Computer Science, Georgia Institute of Technology, Atlanta, Georgia, USA
Dana Randall
  • School of Computer Science, Georgia Institute of Technology, Atlanta, Georgia, USA

Funding

  • Fahrbach, Matthew: Supported in part by an NSF Graduate Research Fellowship under grant DGE-1650044.
  • Randall, Dana: Supported in part by NSF grants CCF-1637031 and CCF-1733812.

Cite AsGet BibTex

Matthew Fahrbach and Dana Randall. Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.37

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Correlated random walk
  • Markov chain Monte Carlo
  • Six-vertex model

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