The hard-core model on random graphs revisited

TL;DR

This paper analyzes the hard-core model on random graphs, reconciling different regimes, and provides conjectures on the density of packings and phase transition nature, linking it to structural glasses and jamming.

Contribution

It offers a closed-form conjecture for the densest packing density on random regular graphs and clarifies the phase transition behavior for large degrees.

Findings

Heuristic cavity method yields a conjecture for maximum packing density.

Phase transition nature is consistent for degrees greater than 16.

Hard-core model acts as a mean-field model for glasses and jamming.

Abstract

We revisit the classical hard-core model, also known as independent set and dual to vertex cover problem, where one puts particles with a first-neighbor hard-core repulsion on the vertices of a random graph. Although the case of random graphs with small and very large average degrees respectively are quite well understood, they yield qualitatively different results and our aim here is to reconciliate these two cases. We revisit results that can be obtained using the (heuristic) cavity method and show that it provides a closed-form conjecture for the exact density of the densest packing on random regular graphs with degree K>=20, and that for K>16 the nature of the phase transition is the same as for large K. This also shows that the hard-code model is the simplest mean-field lattice model for structural glasses and jamming.