Open Access
2011 Three theorems in discrete random geometry
Geoffrey Grimmett
Probab. Surveys 8: 403-441 (2011). DOI: 10.1214/11-PS185


These notes are focused on three recent results in discrete random geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is $\sqrt {2+ \sqrt {2}}$; the proof by the author and Manolescu of the universality of inhomogeneous bond percolation on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on $\mathbb{Z}^2$ is $\sqrt{q}/(1+ \sqrt{q})$. Background information on the relevant random processes is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs.


Download Citation

Geoffrey Grimmett. "Three theorems in discrete random geometry." Probab. Surveys 8 403 - 441, 2011.


Published: 2011
First available in Project Euclid: 30 December 2011

zbMATH: 1245.60093
MathSciNet: MR2861135
Digital Object Identifier: 10.1214/11-PS185

Primary: 60K35
Secondary: 82B43

Keywords: Connective constant , Critical exponent , Ising model , isoradiality , percolation , Random-cluster model , Self-avoiding walk , star–triangle transformation , Universality , Yang–Baxter equation

Rights: Copyright © 2011 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.8 • 2011
Back to Top