Mathematics Colloquia and Seminars

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In the bootstrap percolation model, the sites of an NxN square lattice are initially chosen to be black or white independently with probabilities p or 1-p. The black sites then spread according to the rules: 1. a black site always remains black; 2. a white site at time t becomes black at time t+1 if it has at least two black neighbors. The probability X for the black sites to eventually infect the entire square depends on the square size N and on the initial black site density p. It turns out that when N is large, if p is smaller than roughly C/log(N), X will be very close to 0, and if p is larger than C/log(N), X will be very close to 1. This was proved in 2001 by Alexander Holroyd, who also found that C is equal to exactly pi squared over 18. In the talk I will explain ideas from a joint work with Holroyd and Tom Ligett, in which we found that the constant C is related to an amusing partition identity proved by MacMahon in 1915, which says that the number of integer partitions of an integer N with no parts equal to 1 and no two successive parts is equal to the number of partitions of N into parts all of which are divisible by 2 or 3.