Three theorems in discrete random geometry

Abstract

These notes are focused on three recent results in discrete ran- dom geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is p 2 + √ 2; the proof by the author and Manolescu of the universality of inhomogeneous bond percola- tion on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on Z 2 is √ q/(1 + √ q). Background information on the relevant random pro- cesses is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs. AMS 2000 subject classifications: Primary 60K35; secondary 82B43. Keywords and phrases: Self-avoiding walk, connective constant, per- colation, random-cluster model, Ising model, star-triangle transformation, Yang-Baxter equation, critical exponent, universality, isoradiality.