On the occupancy fraction of the antiferromagnetic Ising model

TL;DR

This paper investigates the extremal occupancy fractions of the antiferromagnetic Ising model on regular graphs, identifying the computational threshold for sampling at certain magnetizations, especially for degree 3 graphs.

Contribution

It determines the occupancy fraction thresholds for regular graphs with degree 3, advancing understanding of sampling complexity in the antiferromagnetic Ising model.

Findings

Thresholds for occupancy fractions are established for degree 3 graphs.

The results cover nearly the entire relevant parameter range for Δ=3.

Extends understanding of computational complexity in Ising model sampling.

Abstract

We study the maximum and minimum occupancy fraction of the antiferromagnetic Ising model in regular graphs. The minimizing problem is known to determine a computational threshold in the complexity of approximately sampling from the Ising model at a given magnetization, and our results determine this threshold for nearly the entire relevant parameter range in the case $\Delta=3$. A small part of the parameter range lies outside the reach of our methods, and it seems challenging to extend our techniques to larger $\Delta$.