The incipient infinite cluster and moments of local quantities in high dimensional percolation
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435
ENS de Lyon
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.