Percolation on Self-Dual Polygon Configurations

In this talk I shall sketch some results Oliver Riordan of Oxford and I have obtained on the critical probabilities in percolation. The story starts with Scullard and Zi, who recently pointed out that a broad class of planar percolation models are self-dual. They stated that in a variety of classes of models depending on a parameter, the parameter giving self-duality is the critical value for percolation. However, noticing self-duality is simply the starting point: the mathematical difficulty is precisely showing that self-duality implies criticality. Riordan and I have managed to overcome this difficulty: we have shown that, for a generalization of the models considered by Scullard and Zi, self-duality indeed implies criticality.

Speaker Details

Bela Bollobas is a Senior Research Fellow at Trinity College, Cambridge, UK and also holds the J. Hardin Chair of Excellence at the University of Memphis. He has proved numerous important results on extremal graph theory, functional analysis, the theory of random graphs, graph polynomials and percolation. In particular, he was the first to prove detailed results about the phase transition in the evolution of random graphs. In addition to over 400 research papers on mathematics, he has written several celebrated books, including “Extremal Graph Theory”, “Random Graphs” ,”Percolation” (with Oliver Riordan), “Modern Graph Theory”, “Combinatorics”, and “Linear Analysis”. Bela has guided dozens of PhD students, including the Fields medalist William Gowers.

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Bela Bollobas
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