Modul: MAT971 Stochastische Prozesse

## An unbounded nodal surface for 3D analytical Gaussian functions

Talk by Dr. Hugo Vanneuville

**Date:** 13.04.22 **Time:** 17.15 - 18.15 **Room:** Y27H12

Let f be the 3D Bargmann-Fock field, which is a Gaussian random analytic function from \R^3 to \R. We prove that a.s. there is an unbounded component in the nodal surface {f=0}. Since 0 is the percolation critical level for f restricted to the plane, this result is equivalent to the fact that the critical level strictly increases between 2D and 3D. This is a classical result for Bernoulli percolation. However, the classical proofs strongly rely on finite energy properties, which are not available for analytical functions. We propose a different approach, that is based on the continuous nature of the model.