Activities This Week

The roughening transition is a conjectured second transition point for 3d percolation, related to surface tension and fluctuations. We will survey the conjecture and the motivation behind it. Time permitting, we will discuss some new results with Hugo Duminil-Copin, Sébastien Ott and Florian Schweiger; and with Zhenhao Cai. All terms will be explained in the talk.

In the `80 Diaconis and others have observed that some naturally occurring Markov chains exhibit a cutoff phenomenon: the distance from the stationary distribution drops from almost maximal to almost zero over a short period of time. It is conjectured that all transitive expander graphs exhibit this phenomenon, but so far very few examples are known. I will survey and explain some results on cutoff for walks on Ramanujan graphs and complexes, and their relation to expansion in groups.