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On the occupancy fraction of the antiferromagnetic Ising model

Ewan Davies, Olivia LeBlanc

TL;DR

This work analyzes the occupancy fraction $\alpha_G(B,\lambda)$ of the antiferromagnetic Ising model on $\Delta$-regular graphs to locate computational thresholds for magnetization-constrained counting. The authors develop a computer-assisted occupancy method based on local views and LP dual feasibility, enabling exact certificates of extremal graphs in restricted parameter regions. They prove that for $\Delta=3$ and $0<B<0.3128$, the complete graph $K_4$ minimizes $\alpha_G(B,\lambda_c(3,B))$, and they show that the complete bipartite graph $K_{3,3}$ maximizes $\alpha_G(B,\lambda)$ in certain regions, with partial symbolic results guiding potential extension to larger $\Delta$. The results illustrate the power and limits of LP-based extremal analysis in constrained spin systems and point to future work needed to scale to higher degrees and deeper local structure.

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 $Δ=3$. A small part of the parameter range lies outside the reach of our methods, and it seems challenging to extend our techniques to larger $Δ$.

On the occupancy fraction of the antiferromagnetic Ising model

TL;DR

This work analyzes the occupancy fraction of the antiferromagnetic Ising model on -regular graphs to locate computational thresholds for magnetization-constrained counting. The authors develop a computer-assisted occupancy method based on local views and LP dual feasibility, enabling exact certificates of extremal graphs in restricted parameter regions. They prove that for and , the complete graph minimizes , and they show that the complete bipartite graph maximizes in certain regions, with partial symbolic results guiding potential extension to larger . The results illustrate the power and limits of LP-based extremal analysis in constrained spin systems and point to future work needed to scale to higher degrees and deeper local structure.

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 . A small part of the parameter range lies outside the reach of our methods, and it seems challenging to extend our techniques to larger .

Paper Structure

This paper contains 8 sections, 4 theorems, 17 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Proof sketch
  4. Proof
  5. Minimization: Theorem \ref{['thm:d3lc']} and Theorem \ref{['thm:d3min']}
  6. Maximization: Theorem \ref{['thm:d3max']}
  7. Discussion and extensions of the method
  8. Running our code

Key Result

Theorem 1

For $\Delta\ge 3$ and $0<B<B_c(\Delta)$, we write $\alpha_c = \alpha_{\inf}(\Delta, B,\lambda_c(\Delta,B))$. Then $\alpha_c$ is a computational threshold in the following sense.

Figures (4)

  • Figure 1: Minimizing the occupancy fraction over 3-regular graphs. The blue region is union of the regions in Theorem \ref{['thm:d3min']} where we proved that $K_4$ minimizes the occupancy fraction. The yellow region is where we numerically evaluated that it is theoretically possible for our method to show that $K_4$ minimizes the occupancy fraction, but we did not push the symbolic proofs of dual feasibility to the limit. The black line is $\lambda_c(3,B)$.
  • Figure 2: Considering only the pictured graphs, $K_4$, the Goose graph, and the Petersen graph respectively, the plot shows the region where each of them has the smallest occupancy fraction out of the three. The black line is $\lambda_c(3,B)$. This shows that it is not possible to strengthen the relevant part of Theorem \ref{['thm:d3min']} by enlarging the regions to cover the entire parameter range $[0,1]^2$, but it does not disprove the original conjecture in DP23 about $K_4$ minimizing when $\lambda=\lambda_c(3,B)$.
  • Figure 3: Maximizing the occupancy fraction over 3-regular graphs. The blue region is union of the regions in Theorem \ref{['thm:d3max']} where we proved that $K_{3,3}$ maximizes the occupancy fraction. The yellow region is where we numerically evaluated that it is theoretically possible for our method to show that $K_{3,3}$ maximizes the occupancy fraction, but we did not push the symbolic proofs of dual feasibility to the limit.
  • Figure 4: Local views one can obtain in $K_{3,3}$. The central vertex is the root $u$ of the local view, and vertices assigned $+$ or $-$ are colored red or blue respectively.

Theorems & Definitions (6)

  • Theorem 1: Davies and Perkins DP23
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture 5
  • Definition 1