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

TL;DR

This paper analyzes the asymptotic behavior of spectral projectors and Gaussian fields on manifolds, revealing anomalies in local Weyl laws and their implications for the topology of random zero sets at critical dimensions.

Contribution

It provides new asymptotic formulas for spectral kernels and applies them to study the topology of random fields, uncovering anomalies in local Weyl laws at critical dimensions.

Findings

Number of zero set components concentrates around aL^{n/m} for n>ms.

Expected Betti numbers grow logarithmically when n=ms.

Explicit constants are derived for specific cases like surfaces with Laplacian.

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 $U_L=\bigoplus_{\lambda\leq L}Ker(A-\lambda Id)$ 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 $\Pi_LA^{-s}$ 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 $n\geq ms$ and consider the family $(\phi_L)_{L>0}$ of smooth random fields on $\mathcal{M}$ defined by \[\phi_L=\sum_{0<\lambda_j\leq L}\lambda_j^{-\frac{s}{2}}\xi_je_j\] for each $L>0$. It turns out that the covariance function of $\phi_L$ is $K_L$. Using this information, we apply the derived asymptotics to study the zero set of $\phi_L$. If $n>ms$ then the number of components of the zero set of $\phi_L$ concentrates around $aL^{\frac{n}{m}}$ for some positive constant $a$. On the other hand, if $n=ms$, each Betti number of the zero set has an expectation bounded by $C\ln\Big(L^{\frac{1}{m}}\Big)^{-\frac{1}{2}}L^{\frac{n}{m}}$ 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 $\frac{1}{4\pi^2}\sqrt{\frac{3}{2}}Vol(\mathcal{M})$.