Anomalies in local Weyl laws and applications to random topology at critical dimension
Résumé
Let M be a smooth manifold of positive dimension n equipped with a smooth density dµ M. Let A be a polyhomogeneous elliptic pseudo-differential operator of positive order m on M which is symmetric for the L 2 scalar product defined by dµ M. For each L > 0, the space U L = λ≤L Ker(A − λId) is a finite dimensional subspace of C ∞ (M). Let Π L be the spectral projector onto U L. Given s ∈ R, we compute the asymptotics of the integral kernel K L of Π L A −s in the cases where n > ms and n = ms respectively. Next, assuming that M is closed, let (e n) n∈N and (λ n) n∈N be the sequence of L 2 normalized eigenfunctions and eigenvalues of A where the latter sequence organized in increasing order. Let (ξ n) 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 (φ L) L>0 of smooth random fields on M defined by φ L = 0<λj ≤L λ − s 2 j ξ j e j for each L > 0. It turns out that the covariance function of φ L is K L. Using this information, we apply the derived asymptotics to study the zero set of φ L. If n > ms then the number of components of the zero set of φ L 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 C ln 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 dµ M is the Riemmanian volume, C equals 1 4π 2 3 2 V ol(M).
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