Intrinsic geometry of critical discrete structures

Motivated by the presence of empirical data on a wide array of real-world networks, there has been an explosion in the number of random graph models proposed to explain various phenomenon observed in real-world systems including power law degree distribution and small world phenomenon. A major general theme in this area, since the time of Erdos and Renyi, has been understanding the properties of components in the critical regime. In the past five years, significant progress has been made in establishing scaling limits of critical random graphs and various constructs on random graphs when they are viewed as metric measure spaces, and understanding the general universality principles underlying such scaling limits. Further, striking connections between these questions and some of the central models of stochastic coalescence, and random interlacements and vacant set left by random walks have emerged. In this talk, we survey some recent results in this area and discuss general methodology for proving such results.