Spatial Mixing and Deterministic Approximate Counting of Multi-spin Systems beyond Bounded Degree Graphs

2026-09-10

AI Summary

本文开发了一种用于超越有界度图的多自旋系统确定性近似计数框架，利用线性分数规划获得边际比率认证界限，并在特定条件下实现完全多项式时间近似方案。

Abstract

We develop a framework for deterministic approximate counting of multi-spin systems beyond bounded-degree graphs. The algorithm recursively constructs rational polytopes containing the true marginal vectors and uses linear-fractional programming to obtain certified bounds on marginal ratios. For positive interactions on graphs of polynomial connective constant $D$, we establish strong spatial mixing and a fully polynomial-time approximation scheme (\textbf{FPTAS}) whenever $Dc<1$, where $c$ bounds the Birkhoff contraction coefficients of the interactions. We further extend the framework to proper colorings of sparse Erdős-Rényi random graphs using recursion on permissive blocks. For every fixed $η\in(0,1)$, sufficiently large fixed $d$, and fixed integer $q\ge(2+η)d$, we obtain an \textbf{FPTAS} for counting proper $q$-colorings of $G\sim\mathcal G(n,d/n)$ with high probability over $G$. This improves the leading constant $3$ in the earlier counting guarantee of Yin and Zhang (APPROX/RANDOM, 2016) to $2$, and asymptotically matches the spatial mixing regime established by Yin (ICALP, 2014).

Problem

• deterministic approximate counting
• multi-spin systems
• spatial mixing
• Erdős-Rényi random graphs
• proper colorings

Innovation

• deterministic approximate counting
• multi-spin systems
• spatial mixing
• linear-fractional programming
• FPTAS