  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.
