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Sampling from the antiferromagnetic Ising model on bipartite, regular expander graphs

Anna Geisler, Mihyun Kang, Michail Sarantis, Ronen Wdowinski

Abstract

The antiferromagnetic Ising model samples subsets of vertices of a graph with weight decaying exponentially in the number of edges induced. We study the problem of sampling from this model on the class of bipartite, regular graphs with good vertex expansion. We show that a natural sampler, namely the Glauber dynamics, mixes exponentially slowly in a wide range of parameters. On the other hand, we give an efficient alternative algorithm for sampling from the Ising model and an FPTAS for its partition function, using polymer models and the cluster expansion method.

Sampling from the antiferromagnetic Ising model on bipartite, regular expander graphs

Abstract

The antiferromagnetic Ising model samples subsets of vertices of a graph with weight decaying exponentially in the number of edges induced. We study the problem of sampling from this model on the class of bipartite, regular graphs with good vertex expansion. We show that a natural sampler, namely the Glauber dynamics, mixes exponentially slowly in a wide range of parameters. On the other hand, we give an efficient alternative algorithm for sampling from the Ising model and an FPTAS for its partition function, using polymer models and the cluster expansion method.
Paper Structure (23 sections, 16 theorems, 123 equations, 2 algorithms)

This paper contains 23 sections, 16 theorems, 123 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Approximation Algorithms
  4. Main results
  5. Structure of the paper
  6. Preliminaries
  7. Definitions and basic facts
  8. Computational aspects
  9. Markov chains and mixing times
  10. Polymer models and the cluster expansion
  11. Slow mixing of the Glauber dynamics
  12. Balanced sets as bottlenecks
  13. The contribution of small sets -- proof of \ref{['lem:small']}
  14. The contribution of large sets -- container lemma and proof of \ref{['lem:middle']}
  15. The contribution of remaining sets -- proof of \ref{['lem:rest']}
  16. ...and 8 more sections

Key Result

Theorem 1.3

Fix $\Delta_2 \ge 1$, $0 \le \kappa < 2$, and $\lambda_0>0$. Then there exist constants $C_0, C > 0$ such that whenever $\lambda \le \lambda_0$ and $\alpha \coloneqq \lambda(1 - e^{-\beta}) \ge \frac{C_0 \log^{3/2} d}{d^{C_*}}$, the mixing time of the Glauber dynamics $M(G)$ on a graph $G \in \mathc

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1: GaKa2004
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 17 more