Here is another video of a smashing short talk by my dear friend Itai Benjamini with beautiful conjectures proposing an important new step in the connection between percolation and conformal geometry.
Here is the link to Itai’s original paper Percolation and coarse conformal uniformization. It turns out that the missing piece for proving the conjecture is a very interesting (innocent looking) theorem of Russo-Seymour-Welsh type.
This conjecture appears also in a paper Itai wrote with me Around two theorems and a lemma by Lucio Russo. Our paper is part of a special issue of Mathematics and Mechanics of Complex Systems (M&MoCS), in honor of Lucio Russo, in occasion of his retirementis. In addition to the conjectural Russo-Seymour-Welsh type theorems, we also present some developments, connections, and problems related to Russo’s lemma and Russo’s 0-1 law.
Lucio Russo
Combinatorics and more 