Log-scale equidistribution of nodal sets in Grauert tubes

Let M_τ0 be the Grauert tube (of some fixed radius τ₀) of a compact, negatively curved, real analytic Riemannian manifold M without boundary. Let φ_λ be a Laplacian eigenfunction on M of eigenvalue −λ² and let φ_λ ^C be its holomorphic extension to M_τ0 . In this article, we prove that on M_τ0 ∖M, there exists a dimensional constant α>0 and a full density subsequence {λ_jk }_k=1 ^∞ of the spectrum for which the masses of the complexified eigenfunctions φ_λjk ^C are asymptotically equidistributed at length scale (log⁡λ_jk )^−α. Moreover, the complex zeros of φ_λjk ^C also become equidistributed on this logarithmic length scale.