Hole probability for nodal sets of the cut-off Gaussian Free Field

Let (Σ,g) be a closed connected surface equipped with a riemannian metric. Let (λn)n∈N and (ψn)n∈N be the increasing sequence of eigenvalues and the sequence of corresponding L2-normalized eigenfunctions of the laplacian on Σ. For each L>0, we consider ϕL=∑0<λn≤Lξnλnψn where the ξn are i.i.d centered gaussians with variance 1. As L→∞, ϕL converges a.s. to the Gaussian Free Field on Σ in the sense of distributions. We first compute the asymptotic behavior of the covariance function for this family of fields as L→∞. We then use this result to obtain the asymptotics of the probability that ϕL is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of ϕL around 12πln⁡L.