Log-scale equidistribution of nodal sets in Grauert tubes

Abstract: Let $M_{\tau_0}$ be the Grauert tube (of some fixed radius $\tau_0$) of a compact, negatively curved, real analytic Riemannian manifold $M$ without boundary. Let $\phi_\lambda$ be a Laplacian eigenfunction on $M$ of eigenvalues $-\lambda<sup>2$ and let $\phi_\lambda<sup>\mathbb{C}$ be its holomorphic extension to $M_{\tau_0}$. In this article, we prove that on $M_{\tau_0} \setminus M$, there exists a dimensional constant $\alpha &gt; 0$ and a full density subsequence ${\lambda_{j_k}}<em>{k=1}<sup>{\infty}$ of the spectrum for which the masses of the complexified eigenfunctions $\phi</em>{\lambda_{j_k}}<sup>\mathbb{C}$ are asymptotically equidistributed at length scale $(\log \lambda_{j_k})<sup>{-\alpha}$. Moreover, the complex zeros of $\phi_{\lambda_{j_k}}<sup>\mathbb{C}$ also become equidistributed on this logarithmic length scale.