Superuniversal Statistics with Topological Origins for non-Hermitian Scattering Singularities
Vortex singularities in speckle patterns formed from random superpositions of waves are an inevitable consequence of destructive interference and are consequently generic and ubiquitous. Singularities are topologically stable, meaning they persist under small perturbations and can only be removed via pairwise annihilation. They have applications including sensing, imaging and energy transfer in multiple fields such as optics, acoustics, and elastic or fluid waves. We generalize the concept of speckle patterns to arbitrary parameter spaces and any complex scalar function that describes wave phenomena involving complicated scattering. In scattering systems specifically, we are often concerned with singularities associated with complex zeros of various functions of the scattering matrix S, such as Coherent Perfect Absorption, Reflectionless Scattering Modes, Transmissionless Scattering Modes, and Exceptional Points. Experimentally, we find that all singularities share a universal statistical property: any quantity that diverges as a simple pole at a singularity has a probability distribution function with a -3 power law tail. The tail of the distribution provides an estimate for the likelihood of finding a given singularity in a generic system. We use these universal statistical results to determine that homogeneous system loss is the most important parameter determining singularity density in a given parameter space of an absorptive scattering system. Finally, we discuss events where distinct singularities coincide in parameter space, which result in higher order singularities that are not topologically protected, and we do not find universal statistical properties for them. We support our empirical results from microwave experiments with Random Matrix Theory simulations and conclude that the statistical results presented hold for all generic non-Hermitian scattering systems.