Effective bounds for monochromatic connectivity measures in two dimensions

Abstract

Abstract We establish numerical lower bounds for the monochromatic connectivity measure in two dimensions introduced by Sarnak and Wigman. This measure dictates among the nodal domains of a random plane wave what proportion have any given number of holes, and how they are nested. Our bounds provide the first effective estimate for the number of simply connected domains and for those that contain a single hole. The deterministic aspect of the proof is to find a single function with a prescribed zero set and, using a quantitative form of the implicit function theorem, to argue that the same configuration occurs in the zero set of any sufficiently close approximation to this function. The probabilistic aspect is to quantify the likelihood of a random wave being close enough to this function.