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

Alejandro Rivera (1)

1 IF - Institut Fourier

Let (Σ, g) be a closed connected surface equipped with a riemannian metric. Let (λ n) n∈N and (ψ n) n∈N be the increasing sequence of eigenvalues and the sequence of corresponding L 2-normalized eigenfunctions of the laplacian on Σ. For each L > 0, we consider φ L = 0<λn≤L ξn √ λn ψ n where the ξ n are i.i.d centered gaussians with variance 1. As L → ∞, φ L 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 φ L is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of φ L around 1 √ 2π ln L.

Domaines: Probabilités [math.PR]; Théorie spectrale [math.SP]

Alejandro Rivera. Hole probability for nodal sets of the cut-off Gaussian Free Field. Advances in Mathematics, 2017, 319. ⟨hal-01278683v2⟩