Cambridge University Press, 21.09.2006 - 323 Seiten
Percolation theory was initiated some fifty years ago as a mathematical framework for the study of random physical processes such as flow through a disordered porous medium. It has proved to be a remarkably rich theory, with applications beyond natural phenomena to topics such as network modelling. The aims of this book, first published in 2006, are twofold. First to present classical results in a way that is accessible to non-specialists. Second, to describe results of Smirnov in conformal invariance, and outline the proof that the critical probability for random Voronoi percolation in the plane is 1/2. Throughout, the presentation is streamlined, with elegant and straightforward proofs requiring minimal background in probability and graph theory. Numerous examples illustrate the important concepts and enrich the arguments. All-in-all, it will be an essential purchase for mathematicians, physicists, electrical engineers and computer scientists working in this exciting area.
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4-marked domain Aizenman arcs black horizontal crossing bond percolation boundary Chapter closed component conformal invariance conformal map connected constant contains corresponding critical probability cross-ratio cycle deﬁned deﬁnition diﬀerent disc discrete domain disjoint edge equivalent exponential decay Figure ﬁnd ﬁnite ﬁrst ﬁxed follows function graph distance Harris’s Lemma Hence hexagonal lattice holds implies independently with probability inequality inﬁnite open cluster inﬁnite open path joining Kesten least line segment Menshikov’s Theorem Note obtained open crossing open horizontal crossing open independently open with probability percolation measure percolation model planar lattice plane points Poisson process probability measure proof of Theorem proved radius random graphs random Voronoi percolation rectangle result Rn(x sequence set of sites site percolation site x Smirnov’s square lattice subset symmetric triangular lattice unoriented upper bound vertex set vertices Voronoi cells Voronoi tessellation weighted graph write