An FPRAS for Antiferromagnetic Ising Models on Random Regular Bipartite Graphs

2026-08-19

AI Summary

本文设计了一种随机近似方案，用于求解随机正则二分图上反铁磁Ising模型的配分函数问题，通过截断和Gibbs分布采样方法实现。

Abstract

We design randomized approximation schemes for the partition function of antiferromagnetic Ising models with uniform external field on random regular bipartite graphs. Our algorithm generalizes the approach of Kocurek, Oveis Gharan and Tjowasi (arXiv, 2026) for hard-core models on the same random graph model beyond the uniqueness threshold. We show that, as long as $λ$ is upper bounded by a constant and $λ(1 - β) \lesssim Δ^{-1/2}$, an efficient randomized algorithm approximates the partition function with high probability. The algorithm first truncates configurations that are large on either side of the bipartition and then samples from Gibbs distributions conditioned on fixed sizes on one or both sides. To choose an optimal truncation bound, we establish concentration properties of the Gibbs distribution on random regular bipartite graphs. Then we apply high-dimensional expansion and prove trickle-down theorems to obtain fast samplers for the conditioned distributions.

Problem

• antiferromagnetic Ising models
• partition function
• random regular bipartite graphs

Innovation

• FPRAS
• antiferromagnetic Ising model
• random regular bipartite graphs
• Gibbs distribution
• high-dimensional expansion