Philippe Sosoe: The incipient infinite cluster and moments of local quantities in in high dimensional percolation

Abstract:

In critical Bernoulli percolation on the lattice, the cluster of the origin is expected to be almost surely finite, but its mean size is infinite. This raises the question of whether a critical percolation process with the cluster of the origin conditioned to be infinite can be defined.

Such a process was constructed in dimension 2 by Kesten, and in various versions in sufficiently high dimensions by van der Hofstad-Jarai and later Heydenreich-van der Hofstad-Hulshof. These constructions use a diagrammatic expansion called the lace expansion. I will explain a new, more general construction of the IIC which circumvents the need for the lace expansion and gives an unconditional result. Time permitting, I will explain how this result is used to compute the asymptotics of moments of quantities like the distance between the origin and a distant point.

Joint work with P Chinmay, J Hanson and S Chatterjee.