Decay of Correlations for the Hardcore Model in Random Regular Graphs

Gibbs measures on trees play a central role in the theory of spins systems on random graphs. We determine the local weak limit of the hardcore model on random regular graphs upto a density for the largest independent set that is bounded by and goes asymptotically to the condensation threshold. We show that the hardcore measure converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence, the reconstruction threshold on the random graph is equal to the reconstruction threshold on the d-regular tree. We expect that our methods should generalize to a wide range of spin systems for which the second moment method holds.

This is joint work with Allan Sly and Prasad Tetali.