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 $\frac{ 1}{ 4\pi^2}\sqrt{ \frac{ 3}{ 2}}Vol(\mathcal{ M})$.
Categories: math.SP math.PR