Zero-free regions and concentration inequalities for hypergraph colorings in the local lemma regime

2026-01-20

AI Summary

This study investigates the distribution of zeros of the partition function in high-dimensional hypergraph coloring and its implications for the statistical properties of random colorings. By introducing an external field to construct a constraint satisfaction problem (CSP) partition function, the authors establish a zero-free region in the complex plane and, for the first time, apply the projection–lifting technique to the complex analysis of CSPs. Combining information percolation with the local lemma, they derive Berry–Esseen-type and Chebyshev-type concentration inequalities for color class sizes. Under the conditions \(k \geq 50\) and \(q \geq 700\Delta^{5/(k-10)}\), they prove the existence of a zero-free strip around \([0,1]\), establishing asymptotic normality and yielding a deterministic approximation algorithm. This work extends the Lee–Yang and Fisher zero theories to hypergraph coloring, offering a novel analytical framework for high-dimensional constraint satisfaction problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web data visualization

Abstract

We show that for $q$-colorings in $k$-uniform hypergraphs with maximum degree $\Delta$, if $k\ge 50$ and $q\ge 700\Delta^{\frac{5}{k-10}}$, there is a"Lee-Yang"zero-free strip around the interval $[0,1]$ of the partition function, which includes the special case of uniform enumeration of hypergraph colorings. As an immediate consequence, we obtain Berry-Esseen type inequalities for hypergraph $q$-colorings under such conditions, demonstrating the asymptotic normality for the size of any color class in a uniformly random coloring. Our framework also extends to the study of"Fisher zeros", leading to deterministic algorithms for approximating the partition function in the zero-free region. Our approach is based on extending the recent work of [Liu, Wang, Yin, Yu, STOC 2025] to general constraint satisfaction problems (CSP). We focus on partition functions defined for CSPs by introducing external fields to the variables. A key component in our approach is a projection-lifting scheme, which enables us to essentially lift information percolation type analysis for Markov chains from the real line to the complex plane. Last but not least, we also show a Chebyshev-type inequality under the sampling LLL condition for atomic CSPs.

Problem

• zero-free regions
• hypergraph colorings
• concentration inequalities
• partition function
• local lemma

Innovation

• zero-free region
• hypergraph coloring
• projection-lifting scheme
• information percolation
• partition function approximation