## Bootstrap Percolation in d Dimensions

- Date: 04/19/2006

Alexander Holroyd (University of British Columbia)

University of British Columbia

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.

Probability Seminar 2006