On the occupancy fraction of the antiferromagnetic Ising model

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 $\Delta$.