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

Abstract

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.

Expected number of nodal components for cut‐off fractional Gaussian fields

“…The function ϕ i is bounded and continuous on K * ⊗ K * except on the set D ⊂ K * ⊗ K * of (x, η x ) such that said bilinear form is degenerate. Let (ϕ l ) l∈N be a sequence of functions in C ∞ c (K * ⊗ K * ) taking values in [0,1] converging pointwise to ϕ i on the complement of D. We study (14) and (15) with ϕ = ϕ l and claim that as l → +∞ we obtain the same statement with ϕ = ϕ i . Let us first check that (14).…”

“…Let (ϕ l ) l∈N be a sequence of functions in C ∞ c (K * ⊗ K * ) taking values in [0,1] converging pointwise to ϕ i on the complement of D. We study (14) and (15) with ϕ = ϕ l and claim that as l → +∞ we obtain the same statement with ϕ = ϕ i . Let us first check that (14). Note that applying (14) with ϕ = 1 we deduce that the random variables…”

“…Let us first check that (14). Note that applying (14) with ϕ = 1 we deduce that the random variables…”

“…As a result, the number of connected components in a fixed compact set is of order . In contrast, the recent work introduced a model of random linear combinations of eigenfunctions of the Laplacian on a closed surface that did not have ‘local limits’. A natural problem is to determine the rate of growth of the number of nodal domains for this new model.…”

“…Remark In the case where , the field is the cut‐off Gaussian Free Field introduced in . By construction, this field converges in distribution to the Gaussian Free Field ( see ) .…”