Around two theorems and a lemma by Lucio Russo
Itaï Benjamini, Gil Kalai · Mathematics and Mechanics of Complex Systems · 2018
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.