Around Two Theorems And A Lemma By Lucio Russo

We describe two directions of study following early work of Lucio Russo. The first direction follows the famous Russo-Seymour-Welsh (RSW) theorem. We describe an RSW-type conjecture by the first author which, if true, would imply a coarse version of conformal invariance for critical planar percolation. The second direction is the study of "Russo's lemma" and "Russo's 0-1 law" for threshold behavior of Boolean functions. We mention results by Friedgut, Bourgain, and Hatami, and present a conjecture by Jeff Kahn and the second author, which may allow applications for finding critical probabilities.