Special Seminar

For the Bargmann--Fock field on $R^d$ with $d > 2$, we prove that the critical level $l_c(d)$ of the percolation model formed by the excursion sets $\{ f \ge l \}$ is strictly positive. This implies that for every l sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case $l=0$), $\{ f = l \}$ contains an unbounded connected component that visits "most" of the ambient space. Our findings (joint work with A. Rivera, PF Rodriguez, and H. Vanneuville) actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. A consequence of this result combined with a result of A. Rivera is that Monochromatic Random Wave in high dimensions percolates almost surely.