The percolation phase transition in the Hamming cube

Speaker  Asaf Nachmias

Host  Yuval Peres

Affiliation  UBC

Duration  01:00:52

Date recorded  30 June 2011

Consider percolation on the Hamming cube 0,1n at the critical probability pc (at which the expected cluster size is 2n/3). It is known that if p=pc(1+O(2-n/3), then the largest component is of size roughly 22n/3 with high probability and that this quantity is non-concentrated. We show that for any sequence eps(n) such that eps(n)2-n/3 and eps(n)=o(1) percolation at pc(1+eps(n)) yields a largest cluster of size (2+o(1))eps(n)2n.

This result settles a conjecture of Borgs, Chayes, van der Hofstad, Slade and Spencer.

Joint work with Remco van der Hofstad.

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