Geometry, Structure and Randomness in Combinatorics

Extremal and Probabilistic Results on Bootstrap Percolation

speaker: Béla Bollobás (DPMMS, Centre for Mathematical Sciences, Cambridge)

abstract: Bootstrap percolation is one of the simplest cellular automata. One of its basic variants is bootstrap percolation on a grid $$[n]^d$$ or lattice $$\mathbb{Z}^d$$ with infection parameter r. This starts with a set $$A_0$$ of initially infected sites (vertices); at each time step $$t = 1, 2, \ldots$$ every site with at least $$r$$ infected neighbours becomes infected. An infected site remain infected forever. We say that $$A_0$$ percolates (with parameter $$r$$) if eventually all sites get infected; the time of percolation is the minimal $$t$$ with $$A_t = V (G)$$, and the speed of percolation is the number of vertices divided by this time.
In the talk I shall present a number of fundamental extremal and probabilistic results on bootstrap percolation, including the fundamen- tal results due to Aizenman, Lebowitz, Schonman, Holroyd, Benevides, Przykucki, and others, and some of the results I have proved recently with Balogh, Morris, Duminil-Copin, Riordan, Holmgren, Smith and Uzzell.

timetable:
Thu 6 Sep, 10:30 - 11:30, Aula Dini
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