Probability and ergodic theory (PET) Ical Atom

The subspace Sub(G) of all subgroups of a countable group G admits a natural structure of a compact metrizable space called the Chabauty space of G. What does the topological structure of the Chabauty space tell us about the algebraic structure of the group G?

A subgroup of Sub(G) is called isolated if it corresponds to an isolated subgroup of G. Isolated subgroups are very special from an algebraic point of view. A group G is called solitary if the isolated points are dense in Sub(G). I will show how the solitary condition is reflected in a surprising way in the permutation representation theory of G. And show how for finitely generated groups the notion of solitary groups generalizes the notion of LERF (subgroup separable) groups.

The talk is based on a joint work with Daniel Kitroser and Jullien Melleray.

I will discuss a conjecture of Benjamini & Schramm from 1996: Any Cayley graph has a non-trivial critical point for percolation (i.e. p_c<1) unless the underlying group is a I finite extension of Z.

I will try to present a strategy to prove this conjecture (in fact some stronger form of it), that involves the notion of EIT = exponential intersection tail measures. Hopefully, all the notions involved (percolation, the critical point p_c, EIT, etc.) will be explained. The aim is to learn these notions and perhaps discuss the weakness or plausibility of the strategy proposed.

Let X and Y be two unbounded positive independent random variables. Write Min_m for the probability of the event {min(X,Y) > m} and Mean_m for that of the event {(X+Y)/2 > m}. We show that the limit inferior of Min_m / Mean_m is always 0 (as m approaches infinity), regardless of the distributions of X and Y. We view this statement as a universal anti-concentration result, and discuss several implications. The proof is elementary but involved, relying on comparison to the “nearest” log-concave measure. We also provide a multiple-variables, weighted variant of this result in the i.i.d. case and pose a conjectured general result encompassing this phenomenon. Joint work with Ohad Feldheim