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Progress in high-dimensional percolation

Banff International Research Station for Mathematical Innovation and Discovery

A major breakthrough in percolation was the 1990 result by Hara and Slade proving mean-field behavior of percolation in high-dimensions, showing that at criticality there is no percolation and identifying several percolation critical exponents. The main technique used is the lace expansion, a perturbation technique that allowed Hara and Slade to compare percolation paths to random walks based on the idea that faraway pieces of percolation paths are almost independent in high dimensions. In this talk, we describe these seminal 1990 results, as well as a number of novel results for high-dimensional percolation that have been derived since and that build on the shoulders of these giants. Time permitting, I intend to highlight the following topics: (1) Critical percolation on the tree and critical branching random walk to fix ideas and to obtain insight in the kind of results that can be proved in high-dimensional percolation; (2) The recent computer-assisted proof, with Robert Fitzner, that identifies the critical behavior of nearest-neighbor percolation above 11 dimensions using the so-called Non-Backtracking Lace Expansion (NoBLE) that builds on the unpublished work by Hara and Slade proving mean-field behavior above 18 dimension; (3) The identification of arm exponents in high-dimensional percolation in two works by Asaf Nachmias and Gady Kozma, using a clever and novel difference inequality argument, and its implications for the incipient infinite cluster and random walks on them; (4) Super-process limits of large critical percolation clusters and the incipient infinite cluster. We assume no prior knowledge about percolation.

Mathematics; Statistical mechanics, structure of matter; Probability theory and stochastic processes; Probability / statistical mechanics

video/mp4

Author affiliation: TU Eindhoven

2017-01-31

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60100

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/