PERCOLATION OF THE EXCURSION SETS OF PLANAR SYMMETRIC SHOT NOISE FIELDS

We prove the existence of a sharp phase transition in the global connectivity of the excursion sets of planar symmetric shot noise fields, with the zero level critical. Our results hold for a wide class of mark distributions, including the Gaussian, Uniform, and Rademacher cases. Our main assumption on the shot noise kernel is that it is positive, symmetric, and has sufficient tail decay (depending on the mark distribution); for example, for Gaussian and Uniform marks we require polynomial decay with exponent at least three, whereas for Rademacher marks we require super-polynomial decay