Real and Complex Zeros of Riemannian Random Waves

We show that the expected limit distribution of the real zero set of a Gaussian random linear combination of eigenfunctions with frequencies from a short interval (‘asymptotically fixed frequency’) is uniform with respect to the volume form of a compact Riemannian manifold (M, g). We further show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauert tube in the complexification of M tends to a limit current. This article is concerned with the real and complex zero sets of Riemannian random waves on a real analytic Riemannian manifold (M, g). To define Riemannian random waves, we fix an orthonormal basis {φλj} of real-valued eigenfunctions of the Laplacian Δg of (M, g), Δgφλj = λ 2 jφλj , 〈φλj , φλk〉 = δjk, and define Gaussian ensembles of random functions f = ∑ j cjφλj of the following two types: • The asymptotically fixed frequency ensemble HIλ , where Iλ = [λ, λ + 1] and where HIλ is the vector space of linear combinations fλ = ∑ j:λj∈[λ,λ+1] cj φλj , (1) of eigenfunctions with λj (the frequency) in an interval [λ, λ + 1] of fixed width. (Note that it is the square root of the eigenvalue of Δ, not the eigenvalue, which is asymptotically fixed). • The cut-off ensembles H[0,λ] where the frequency is cut-off at λ: fλ = ∑ j:λj≤λ cj φλj , (2) By random, we mean that the coefficients cj are independent Gaussian random variables with mean zero and with the variance defined so that the expected L norm of f equals one. Equivalently, the real vector spaces H[0,λ], resp. HIλ are endowed with the inner product 〈u, v〉 = ∫ M uvdVg (where dVg is the volume form of (M, g)) and random means that we equip the vector spaces with the induced Gaussian measure. Our main results given the asympototic distribution of real and complex zeros of such Riemannian random waves in the high frequency limit λ → ∞. The real zeros are straightforward to define. For each fλ ∈ H[0,λ] or HIλ we associated to the zero set Zfλ = {x ∈ M : fλ(x) = 0} the positive measure 〈|Zfλ|, ψ〉 = ∫