An $\widetilde{O}\left(n^3 \right)$-Time Sampler for Zero-Field Ferromagnetic Ising Models

2026-08-13

AI Summary

本文提出了一种在任意图上近似采样无外场铁磁Ising模型的方法，通过结合切片稀疏化和新的混合时间分析，在O(n^2)时间内解决该问题。

Abstract

We give an approximate sampler for ferromagnetic Ising models with no field on arbitrary graphs that runs in time $\widetilde O(m+n)+\widetilde O_\beta\left(n^3\log^3\frac{1}{\epsilon}\right)$, where $n$ and $m$ are the numbers of vertices and edges, respectively, and $\epsilon$ is the approximation error. Our algorithm utilises the machinery behind the classic cut sparsifier by Bencz\'ur and Karger (2015) and the transport flow idea by Chen, Feng, Ju, Miao, Yin, and Zhang (2025). The main proofs were found using GPT-5.6 Sol Pro.

Problem

• ferromagnetic Ising models
• sampler
• arbitrary graphs
• approximation error

Innovation

• Approximate Sampler
• Ferromagnetic Ising Models
• Mixing Time Analysis
• Monotone Edge-Count Poincaré Inequality