Topologies of nodal sets of random band limited functions

TL;DR

This paper demonstrates that the topologies and nestings of zero and nodal sets of Gaussian band limited functions follow universal distribution laws, with implications for random waves and algebraic hypersurfaces.

Contribution

It establishes universal laws governing the topology and nesting of nodal sets for a broad class of random functions, including monochromatic waves and algebraic hypersurfaces.

Findings

Universal distribution laws for topologies of nodal sets

Qualitative features of the distributions' supports determined

Results apply to random monochromatic waves and algebraic hypersurfaces

Abstract

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.