Vol. 6, No. 2, 2018

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Around two theorems and a lemma by Lucio Russo

Itai Benjamini and Gil Kalai

Vol. 6 (2018), No. 2, 69–75
Abstract

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.

Keywords
percolation, Russo–Seymour–Welsh theorem, Russo's lemma, Russo's 0$\mskip1mu$–1 law, conformal uniformization, discrete isoperimetry
Mathematical Subject Classification 2010
Primary: 60K35, 05C80, 30F10, 68QXX, 82B43
Milestones
Received: 16 January 2017
Revised: 3 January 2018
Accepted: 4 February 2018
Published: 29 May 2018

Communicated by Raffaele Esposito
Authors
Itai Benjamini
Department of Mathematics
Weizmann Institute
Rehovot
Israel
Gil Kalai
Einstein Institute of Mathematics
Hebrew University of Jerusalem
Givat Ram Campus
Jerusalem
Israel