Anomalies in local Weyl laws and applications to random topology at critical dimension

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 onM which is symmetric for the L scalar product defined by dμM. For each L > 0, the space UL = ⊕ λ≤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 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 L 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 onM defined by