H. Vanneuville: Existence of an unbounded nodal surface for 3D smooth Gaussian fields

Abstract: This talk is about the following question: Let f be a smooth (stationary, centered) Gaussian field on \mathbb{R}^d and let u be a real number. Is there an unbounded component in \{f > u\}?

One of the main difficulties in the study of this model is that it is very rigid (in particular, the fields that we will consider satisfy the analytic continuation property, so we do not have any finite energy property).

In the talk, I will review some open problems of the area and then focus on a recent work with Hugo Duminil-Copin, Alejandro Rivera and Pierre-Fran├žois Rodriguez, where we prove that the following holds for a family of fields whose covariance is positive and decays sufficiently fast: Contrary to the planar case, for u=0 (and even u close to 0), a.s. there is an unbounded component in \{f > u\} if d \ge 3. In particular, there is an unbounded component in \{f=0\}.

Actually, since it is known that the critical level in dimension 2 is 0, this result is that the critical level stricly decreases between dimensions 2 and 3. The classical proof for Bernoulli percolation does not seem to extend to this model and we propose a new proof, where we make crucial use of the continuous nature of the model.

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