Quantum Unique Ergodicity and the Asymptotic Distribution of Nodal Domains on Hyperbolic Surfaces: A Transfinite Approach

Mathematical Applications of Science Fiction We present a rigorous derivation of the asymptotic distri-bution of nodal domains for high-energy eigenfunctions ofthe Laplacian on compact hyperbolic surfaces of constantnegative curvature. By invoking the principle of QuantumUnique Ergodicity (QUE) and lifting the nodal measure tothe unit tangent bundle via microlocal analysis, we establisha correspondence between the zero sets of eigenfunctionsand the ergodic flow of the geodesic frame. We introduce aTransfinite Nodal Measure νU ∞ constructed via an ultrafilterlimit of the normalized Hausdorff measures of the nodalsets. We prove that under the assumption of ArithmeticQUE, the nodal domains percolate in a manner consistentwith the Bogomolny-Schmit conjecture, but with correctionsarising from the non-trivial topology of the moduli space ofsemi-classical limits. We provide eight fundamental lemmascharacterizing the Sobolev wavefront sets and the decay ofthe off-diagonal spectral projector kernel.