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Non-existence probabilities and lower tails in the critical regime via Belief Propagation

Matthew Jenssen, Will Perkins, Aditya Potukuchi, Michael Simkin

Abstract

We compute the logarithmic asymptotics of the non-existence probability (and more generally the lower-tail probability) for a wide variety of combinatorial problems for a range of parameters in the `critical regime' between the regime amenable to hypergraph container methods and that amenable to Janson's inequality. Examples include lower tails and non-existence probabilities for subgraphs of random graphs and for $k$-term arithmetic progressions in random sets of integers. Our methods apply in the general framework of estimating the probability that a $p$-random subset of vertices in a $k$-uniform hypergraph induces significantly fewer hyperedges than expected. We show that under some simple structural conditions on the hypergraph and an upper bound on $p$ determined by a phase transition in the hard-core model on the infinite $k$-uniform, $Δ$-regular, linear hypertree, this probability can be accurately approximated by the Bethe free energy evaluated at the unique fixed point of a Belief Propagation operator on the hypergraph.

Non-existence probabilities and lower tails in the critical regime via Belief Propagation

Abstract

We compute the logarithmic asymptotics of the non-existence probability (and more generally the lower-tail probability) for a wide variety of combinatorial problems for a range of parameters in the `critical regime' between the regime amenable to hypergraph container methods and that amenable to Janson's inequality. Examples include lower tails and non-existence probabilities for subgraphs of random graphs and for -term arithmetic progressions in random sets of integers. Our methods apply in the general framework of estimating the probability that a -random subset of vertices in a -uniform hypergraph induces significantly fewer hyperedges than expected. We show that under some simple structural conditions on the hypergraph and an upper bound on determined by a phase transition in the hard-core model on the infinite -uniform, -regular, linear hypertree, this probability can be accurately approximated by the Bethe free energy evaluated at the unique fixed point of a Belief Propagation operator on the hypergraph.

Paper Structure

This paper contains 23 sections, 38 theorems, 265 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $H$-free graphs
  3. Lower tails for subgraphs
  4. Avoiding $k$-APs
  5. Independent and sparse sets in hypergraphs
  6. Non-regular hypergraphs
  7. Methods
  8. Outline
  9. Preliminaries: Hypergraphs and Gibbs measures
  10. Hypergraph basics and terminology
  11. Gibbs measures and partition functions
  12. The Weitz hypertree
  13. Construction of the Weitz hypertree
  14. Weitz tree of asymptotically tree-like hypergraphs
  15. Contraction, Uniqueness, and Fixed Points
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $H$ be a strictly $2$-balanced graph with chromatic number $\chi(H) \ge 3$. Then Similarly, $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: The function $x_{3,1}^*$ from Lemma \ref{['lemkAPfixedpoint']} with $k=3$, $c=1$, representing the density of elements in a random $3$-AP-free subset of $[n]$ chosen with prior density $n^{-1/2}$.
  • Figure 2: A 3-uniform linear hypertree.

Theorems & Definitions (86)

  • Theorem 1.1: janson1987uczakluczak2000trianglebalogh2015independentsaxton2015hypergraph
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Conjecture 1
  • Definition 1
  • Theorem 1.7
  • Theorem 1.8
  • ...and 76 more