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Bootstrap percolation and kinetically constrained spin models: critical lengths and mixing time scales

Presented by: 
Fabio Martinelli Università degli Studi Roma Tre
Wednesday 13th July 2016 - 09:45 to 10:30
INI Seminar Room 1
Co-authors: Cristina Toninelli (Univ Paris VII Diderot), Rob Morris (IMPA)

In recent years, a great deal of progress has been made in understanding the behaviour of a particular class of monotone cellular automata, commonly known as bootstrap percolation. In particular, if one considers only two-dimensional automata, then we now have a fairly precise understanding of the typical evolution of these processes, starting from p-random initial conditions of infected sites. Given a bootstrap model, one can consider the associated kinetically constrained spin model in which the state (infected or healthy) of a vertex is resampled (independently) at rate 1 by tossing a p-coin if it could be infected in the next step by the bootstrap process, and remains in its current state otherwise. Here p is the probability of infection. The main interest in KCM’s stems from the fact that, as p → 0, they mimic some of the most striking features of the glass transition, a major and still largely open problem in condensed matter physics. In this talk, motivated by recent universality results for bootstrap percolation, I’ll discuss some “universality conjectures” concerning the growth of the (random) infection time of the origin in a KCM as p → 0. Joint project with R. Morris (IMPA) and C. Toninelli (Paris VII),
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons