The hard-core model on random graphs revisited

Jean Barbier,Florent Krzakala,Lenka Zdeborová,Pan Zhang

doi:10.1088/1742-6596/473/1/012021

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.

Highlights

The hard-core model is defined as follows: We consider a graph G = (V, E) of size N = |V | and associate an occupation number σi ∈ {0, 1} to every vertex i ∈ V, where 0 stands for free and 1 for occupied
The result is that whereas the current statistical physics picture with continuous phase transition is correct for graphs of small average degree (K < 16 for random regular graphs), the model has a discontinuous phase transition for degree K > 16 and the large K expansion of the corresponding equations is in agreement with all the mathematical results. This is showing that for large dimensional hard spheres this simple hard-core model is a very good lattice model. Another contribution of this paper is to point out that on the K-regular random graphs the one-step replica symmetry broken result for the size of largest independent set, or minimal vertex cover, is stable towards more steps of replica symmetry breaking for K ≥ 20, and for K ≥ 20 we provide a closed form conjecture about the exact values of this density
We matched the two extreme regimes of the hard-core model on random regular graphs to give a picture of the behavior of the solution space for the full range of average degrees

Summary

Continuous clustering transition

When the clustering and condensation transitions coincide they coincide with the local stability of the replica symmetric solution eq (9). In such a case the density ρl marks a point below which the replica symmetric solution is exact. In particular below which the annealed (first moment) calculation counts correctly the number of configurations N (ρ) at a given density ρ in the sense that sRS(ρ) lim. Ρl the number of configurations of a given density is strictly smaller that the replica symmetric result. Note that the local stability of the replica symmetric solution is equivalent to the Kesten-Stigum bound for the reconstruction on graphs [36]

Discontinuous clustering transition and condensation transition

Conclusion