The hard-core model on random graphs revisited

The hard-core model on random graphs revisited

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 reconc...

Random cubic planar graphs revisited

Slow mixing of Glauber Dynamics for the hard-core model on regular bipartite graphs

Let Σ = (V, E) be a finite, d-regular bipartite graph. For any λ > 0 let πλ be the probability measure on the independent sets of Σ in which the set I is chosen with probability proportional to λ|I| (πλ is the hard-core measure with activity λ on Σ). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is πλ. We show that when λ is large enough (as a ...

On the Structure of the Core of Sparse Random Graphs

In this paper, we investigate the structure of the core of a sparse random graph above the critical point. We determine the asymptotic distributions of the total number of isolated cycles there as well as the joint distributions of the isolated cycles of fixed lengths. Furthermore, focusing on its giant component, we determine the asymptotic joint distributions of the cycles of fixed lengths th...

On phase transition in the hard-core model on Z

It is shown that the hard-core model on Zd exhibits a phase transition at activities above some function λ(d) which tends to zero as d→∞; that is: Consider the usual nearest neighbor graph on Zd, and write E and O for the sets of even and odd vertices (defined in the obvious way). Set ΛM = Λ d M = {z ∈ Z : ‖z‖∞ ≤M}, ∂ΛM = {z ∈ Z : ‖z‖∞ = M}, and write I(ΛM ) for the collection of independent se...