A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. 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. We expect that our methods will generalize to a wide range of spin systems for which the second moment method holds.
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