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

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 $\frac{1}{4\pi^2}\sqrt{\frac{3}{2}}Vol(\mathcal{M})$.