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

Abstract

Let $\mathcal{M}$ be a smooth manifold of positive dimension $n$ equipped with a smooth density $d{\mu }_{\mathcal{M}}$. Let $A$ be a polyhomogeneous elliptic pseudo-differential operator of positive order $m$ on $\mathcal{M}$ which is symmetric for the $L^2$ scalar product defined by $d{\mu }_{\mathcal{M}}$. For each $L>0$, the space $U_L={⨁}_{\lambda \le L}Ker(A-\lambda Id)$ is a finite dimensional subspace of $C^{\infty }(\mathcal{M})$. Let ${\Pi }_L$ be the spectral projector onto $U_L$. Given $s\in \mathbb{R}$, we compute the asymptotics of the integral kernel $K_L$ of ${\Pi }_L{A}^{-s}$ in the cases where $n>ms$ and $n=ms$ respectively. Next, assuming that $\mathcal{M}$ is closed, let $(e_n{)}_{n\in \mathbb{N}}$ and $({\lambda }_n{)}_{n\in \mathbb{N}}$ be the sequence of $L^2$ normalized eigenfunctions and eigenvalues of $A$ where the latter sequence organized in increasing order. Let $({\xi }_n{)}_{n\in \mathbb{N}}$ be a sequence of independent centered gaussians of variance $1$. We fix a parameter $s\in \mathbb{R}$ such that $n\ge ms$ and consider the family $({\varphi }_L{)}_{L>0}$ of smooth random fields on $\mathcal{M}$ defined by \[ϕ_L=\sum_{0<λ_j\leq L}λ_j^{-\frac{s}{2}}ξ_je_j\] for each $L>0$. It turns out that the covariance function of ${\varphi }_L$ is $K_L$. Using this information, we apply the derived asymptotics to study the zero set of ${\varphi }_L$. If $n>ms$ then the number of components of the zero set of ${\varphi }_L$ concentrates around $a{L}^{\frac{n}{m}}$ for some positive constant $a$. On the other hand, if $n=ms$, each Betti number of the zero set has an expectation bounded by $C\ln (L^{\frac{1}{m}}{)}^{-\frac{1}{2}}{L}^{\frac{n}{m}}$ where $C$ is an explicit constant. When $\mathcal{M}$ is a closed surface with a Riemmanian metric, $A$ is the Laplacian and $d{\mu }_{\mathcal{M}}$ is the Riemmanian volume, $C$ equals $\frac{1}{4{\pi }^2}\sqrt{\frac{3}{2}}Vol(\mathcal{M})$.

This manuscript was heavily revised and replaced by two new submissions: arXiv:1801.07598, arXiv:1801.06999 . These submissions were made via Hal and intended as a replacement of the present submission. That they appeared as separate from the original is an unfortunate accident

References in corpus (3)

• Random wave functions and percolation
• Universal components of random nodal sets
• C-infinity Scaling Asymptotics for the Spectral Function of the Laplacian