Monday, January 30, 2012

1110.2395 (Geoffrey Grimmett)

Three theorems in discrete random geometry    [PDF]

Geoffrey Grimmett
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 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.
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