Abstract : In this talk we will consider the hard-core model (also known as random independent set) on a general Galton-Watson branching process tree. Given an activity parameter &lambda > 0 and a progeny distribution N we will characterize the phase transition phenomenon in terms of unique solution of a recursive distributional equation (RDE). This will be a generalization of earlier work of Kelly (1985), who looked at the same model on regular trees. Moreover we will show that in the uniqueness domain the long range independence property holds. This will enable us to deduce that the size of a random independent set distributed according to a hard-core model on a sparse random graph (Erdos-Renyi random graph or random regular graph) scales with its size, provided the parameters are such that the associated limiting Galton-Watson tree model is in the uniqueness domain. We will also show that the limiting constant can be (explicitly) computed in terms of the unique solution of the associated RDE.

[The talk will be self contained and no prior knowledge about hard-core model or recursive distributional equations will be assumed.]