2016/11/07 by Alejandro Rivera, Rivera, Alejandro
Mathematics · Physics and Astronomy · #35P20 (Primary) #60G60 (Secondary) #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1611.02018
openalex publication_date 2016/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a smooth manifold of positive dimension n equipped with a smooth density dμM. Let A be a polyhomogeneous elliptic pseudo-differential operator of positive order m on M which is symmetric for the L2 scalar product defined by dμM. For each L>0, the space UL=\bigoplusλ≤ LKer(A-λId) is a finite dimensional subspace of C^∞(M). Let ΠL be the spectral projector onto UL. Given s∈ℝ, we compute the asymptotics of the integral kernel KL of ΠLA-s in the cases where n>ms and n=ms respectively. Next, assuming that M is closed, let (en)n∈ℕ and (λn)n∈ℕ be the sequence of L2 normalized eigenfunctions and eigenvalues of A where the latter sequence organized in increasing order. Let (ξn)n∈ℕ be a sequence of independent centered gaussians of variance 1. We fix a parameter s∈ℝ such that n≥ ms and consider the family (ϕL)L>0 of smooth random fields on M defined by \[ϕL=∑_00. It turns out that the covariance function of ϕL is KL. Using this information, we apply the derived asymptotics to study the zero set of ϕL. If n>ms then the number of components of the zero set of ϕL concentrates around aL(n)/(m) for some positive constant a. On the other hand, if n=ms, each Betti number of the zero set has an expectation bounded by Cln(L(1)/(m))-(1)/(2)L(n)/(m) where C is an explicit constant. When M is a closed surface with a Riemmanian metric, A is the Laplacian and dμM is the Riemmanian volume, C equals (1)/(4π2)√((3)/(2))Vol(M).