Bootstrap Percolation in d Dimensions
Topic
Modified bootstrap percolation is a very simple cellular automaton
model for nucleation and growth. Sites in the cube {1,...,L}^d are
initially occupied independently with probability p. At subsequent
steps, an unoccupied site becomes occupied if it has at one occupied
neighbour in each of the d dimensions. It turns out that for any
p>0, the entire cube becomes occupied (with high probability) if L
is large enough. One can ask how large L needs to be for this to
happen. For d>=2, the answer is (approximately) exp exp ... exp
(lambda/p), where the exponential is iterated d-1 times, and the
constant lambda equals pi^2/6. This is proved by a surprising induction
on the dimension.
The modified model above is a seemingly innocuous variation on the 'standard' bootstrap percolation model, in which a site becomes occupied if it has at least d occupied neighbours. However, the existence of a sharp threshold lambda as above is unproved for the standard model, except in the case d=2, where the threshold occurs at exp(lambda'/p), with lambda'= pi^2/18.
See http://www.math.ubc.ca/~holroyd/boot/ for a picture of the standard model in d=2.
The modified model above is a seemingly innocuous variation on the 'standard' bootstrap percolation model, in which a site becomes occupied if it has at least d occupied neighbours. However, the existence of a sharp threshold lambda as above is unproved for the standard model, except in the case d=2, where the threshold occurs at exp(lambda'/p), with lambda'= pi^2/18.
See http://www.math.ubc.ca/~holroyd/boot/ for a picture of the standard model in d=2.
Speakers
Additional Information
Probability Seminar 2006
Alexander Holroyd (University of British Columbia)
Alexander Holroyd (University of British Columbia)