The percolation phase transition in the Hamming cube


June 30, 2011


Asaf Nachmias




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.


Asaf Nachmias

Asaf is an Assistant Professor in the mathematics department at the University of British Columbia. He received his PhD at UC Berkeley in 2008, spent two years as a postdoc at MSR and one year as a postdoc at MIT.