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

TL;DR

This paper analyzes the asymptotic behavior of non-existence and lower-tail probabilities in combinatorial structures within the critical regime, using Belief Propagation and Bethe free energy approximations.

Contribution

It introduces a new method applying Belief Propagation to estimate probabilities in the critical regime of hypergraph-related problems, bridging existing techniques.

Findings

Accurately approximates non-existence probabilities using Bethe free energy.

Applies to subgraph counts and arithmetic progressions in random sets.

Identifies phase transition conditions for the hypergraph model.

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, $\Delta$-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.