Matchings on Random Regular Hypergraphs

TL;DR

This paper analyzes the monomer-dimer partition function on random regular hypergraphs, establishing free-energy limits and conditions for replica symmetry, with implications for understanding matchings in complex networks.

Contribution

It provides explicit quenched free-energy limits for the model and identifies regimes where the replica-symmetric solution is unique and accurate.

Findings

Convergence of normalized log partition function in certain regimes

Verification of replica-symmetric saddle as the global maximizer

First-moment upper tail estimate for maximum matching size

Abstract

We study the monomer--dimer partition function on the configuration model of random $d$-regular, $l$-uniform hypergraphs. For fixed $d,l\ge2$, we prove quenched free-energy limits in explicit parameter regimes. The proof combines fixed-density first-moment asymptotics, a two-overlap second-moment variational analysis, and a subgraph-conditioning argument for the short cycles of the incidence structure. The main technical point is to identify regimes in which the replica-symmetric saddle is the unique global maximizer of the second-moment rate function. In those regimes the normalized logarithm of the total matching partition function converges in probability to an explicit variational value. We also prove the corresponding result for the weighted partition function whenever the maximizing density lies in the verified replica-symmetric region, give an additional checkable criterion for that region, and record a first-moment upper tail estimate for the maximum matching size.