The Sharp Threshold Phenomenon for Percolation of Gaussian Fields
vendredi 26 octobre 2018, 9h30 - 10h30
Let f be a smooth centered stationary Gaussian field on the plane. Given a real number l, one can color the plane in black where f takes values above -l and in white where it is below -l. One can then ask whether the black set « percolates », meaning that it has an unbounded black cluster. This question is analogous to that of Bernoulli percolation on a planar lattice. As in the latter model, in some cases, the Gaussian model exhibits a « sharp phase transition » at the level l=0. This means first that for all l0 there is a.s. percolation. We present a proof of this result for the case of a specific, natural field on the plane, called the Bargmann-Fock field.