An introduction to Benjamini-Schramm limits in the context of random graphs

I will describe the Erdos-Renyi random graph model. Through the discussion of its Benjamini-Schramm limit, which is a local notion of limit for a sequence of graphs, I will discuss some of its global geometric properties (such as existence of a giant component and average graph distance). I will also mention some recent work (joint with E.Lubetzky, Y.Peres and A.Sly) on the cutoff phenomenon for random walks on such random graphs.