Abstract: In this talk we begin with a brief overview of classical results on phase transitions in Erdos-Renyi random graphs and continue with more recent results on random planar graphs. We will discuss how the planarity constraint imposed on a random graph affects its global and local structure, such as the component structure, the maximum degree and Benjamini-Schramm local weak limit.

Abstract: I will define the double random current representation of the Ising modeland present some of its important applications based on the switching lemma. Afterwards, in the case when the underlying graph is planar, I will talk about a measure preserving map between double random currents and dimers, and will mention several of its consequences.