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Log-scale equidistribution of nodal sets in Grauert tubes

Let $M_{τ_0}$ be the Grauert tube (of some fixed radius $τ_0$) of a compact, negatively curved, real analytic Riemannian manifold $M$ without boundary. Let $ϕ_λ$ be a Laplacian eigenfunction on $M$ of eigenvalues $-λ^2$ and let $ϕ_λ^\mathbb{C}$ be its holomorphic extension to $M_{τ_0}$. In this article, we prove that on $M_{τ_0} \setminus M$, there exists a dimensional constant $α> 0$ and a full density subsequence $ \{λ_{j_k}\}_{k=1}^{\infty}$ of the spectrum for which the masses of the complexified eigenfunctions $ϕ_{λ_{j_k}}^\mathbb{C}$ are asymptotically equidistributed at length scale $(\log λ_{j_k})^{-α}$. Moreover, the complex zeros of $ϕ_{λ_{j_k}}^\mathbb{C}$ also become equidistributed on this logarithmic length scale.