Expected values of eigenfunction periods

Abstract

Let $(M,g)$ be a compact Riemannian surface. Consider a family of $L^2$ normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form $-h_j^2{\Delta }_g{\varphi }_{h_j}={\varphi }_{h_j}$, whose eigenvalues satisfy $h{h}_j^{-1}\in (1,1+hD]$ for $D>0$ a large enough constant. Let ${\mathbf{P}}_h$ be a uniform probability measure on the $L^2$ unit-sphere $S_h$ of this cluster of eigenfunctions and take $u\in S_h$. Given a closed curve $\gamma \subset M$, there exists $C_1(\gamma ,M),C_2(\gamma ,M)>0$ and $h_0>0$ such that for all $h\in (0,h_0],$ \begin{equation*} C_1 h^{1/2} \leq \mathbf{E}_{h} \bigg[ \big| \int_γ u \, d σ\big| \bigg] \leq C_2 h^{1/2} . \end{equation*} This result contrasts the deterministic $\mathcal{O}(1)$ upperbounds obtained by Chen-Sogge \cite{CS}, Reznikov \cite{Rez}, and Zelditch \cite{Zel}. Furthermore, we treat the higher dimensional cases and compute large deviation estimates. Under a measure zero assumption on the periodic geodesics in $S^{\ast }M$, we can consider windows of small width $D=1$ and establish a $\mathcal{O}(h^{1/2})$ estimate. Lastly, we treat probabilistic $L^q$ restriction bounds along curves.

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