Joshua Erde

Percolation in high-dimensional graphs

In their ground-breaking paper which introduced the model of random graphs, Erdos and Renyi showed that the component structure of the binomial random graph changes dramatically near to the critical point p=1/n. A similar phase transition has been observed in a number of related random graph models, where under the right scaling their broad-scale structure near to the percolation threshold quantitatively resembles that of G(n,p). In this talk I will discuss some recent results about the limits of this *universality* phenomenon and its connection to notions of expansion in graphs, in particular in the case of graphs which come with some underlying high-dimensional geometric structure.