Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jingcheng Liu, Yixiao Yu

TL;DR

We address zero-free regions for partition functions of CSP-like systems arising from hypergraph q-colorings under local lemma conditions, proving a Lee–Yang zero-free strip around [0,1] for k≥50 and q≥700Δ^{5/(k-10)}. The core method extends the complex-extension Markov chain framework to CSPs via a projection–lifting scheme, combining a projection to color buckets with a percolation/witness-based analysis to control complex marginals and lift bounds to the original measure. Consequences include Berry–Esseen type CLTs and Local CLTs for color-class sizes under uniform colorings, Chebyshev-type concentration, and a deterministic Barvinok-style FPTAS for the partition function in the zero-free region, with parallel zero-freeness results for CNF formulas and Fisher zeros. Altogether, the work provides a unified analytic framework for zero-freeness, central limit behavior, and deterministic counting in the local lemma regime, with potential applicability to a broad class of CSPs.

Abstract

We show that for $q$-colorings in $k$-uniform hypergraphs with maximum degree $Δ$, if $k\ge 50$ and $q\ge 700Δ^{\frac{5}{k-10}}$, 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.

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

TL;DR

We address zero-free regions for partition functions of CSP-like systems arising from hypergraph q-colorings under local lemma conditions, proving a Lee–Yang zero-free strip around [0,1] for k≥50 and q≥700Δ^{5/(k-10)}. The core method extends the complex-extension Markov chain framework to CSPs via a projection–lifting scheme, combining a projection to color buckets with a percolation/witness-based analysis to control complex marginals and lift bounds to the original measure. Consequences include Berry–Esseen type CLTs and Local CLTs for color-class sizes under uniform colorings, Chebyshev-type concentration, and a deterministic Barvinok-style FPTAS for the partition function in the zero-free region, with parallel zero-freeness results for CNF formulas and Fisher zeros. Altogether, the work provides a unified analytic framework for zero-freeness, central limit behavior, and deterministic counting in the local lemma regime, with potential applicability to a broad class of CSPs.

Abstract

We show that for -colorings in -uniform hypergraphs with maximum degree , if and , there is a "Lee-Yang" zero-free strip around the interval 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 -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.
Paper Structure (43 sections, 63 theorems, 184 equations, 1 figure, 2 algorithms)

This paper contains 43 sections, 63 theorems, 184 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Technical overview
  4. Zero-freeness via constraint-wise self-reduction
  5. Projection and lifting scheme
  6. Analyzing the projected measure via complex Glauber dynamics
  7. CLTs, LCLTs and concentration inequalities under LLL-like conditions
  8. Preliminaries
  9. Complex normalized measures
  10. Graphical models and their conditional measure
  11. Glauber dynamics: random scan and systematic scan
  12. CSP formulas
  13. Lovász local lemma
  14. $2$-tree
  15. Controlling marginal measures via analytic percolation
  16. ...and 28 more sections

Key Result

Theorem 1.1

Fix any integers $k\ge 50$ and $q\ge 700\Delta^{\frac{5}{k-10}}$. Let $H=(V, \mathcal{E})$ be a $k$-uniform hypergraph with maximum degree $\Delta$. Then $Z^{\mathrm{co}}_H(\lambda)\neq 0$ for any $\lambda\in \mathbb{C}$ satisfying $\exists \lambda_c\in[0,1]$, such that $|\lambda - \lambda_c|\le \fr

Figures (1)

  • Figure 1: Each displayed color represents one color bucket in the state-compression scheme. The two leftmost hyperedges form a bad cluster since their vertices belong to the same color bucket.

Theorems & Definitions (134)

  • Theorem 1.1
  • Theorem 1.2: name=CLT for a color class in hypergraph $q$-coloring
  • Theorem 1.3: name=Local CLT for a color class in hypergraph $q$-coloring
  • Theorem 1.4: Chebyshev type inequality for atomic CSP formulas
  • Theorem 2.1: erdHos1975problems
  • Theorem 2.2: haeupler2011new
  • Theorem 2.3: moser2010constructive
  • Definition 2.4: $2$-tree
  • Definition 2.5: Construction of a maximal $2$-tree Vishesh21towards
  • Lemma 2.6: borgs2013left,feng2021rapid
  • ...and 124 more