Abstract:

Random eigenfunctions of Lapalcian provide a model for "typical" high-energy eigenfunctions of the Laplace operator in generic domains. In this talk we will discuss random plane waves that could be thought of as random linear combination of all possible plane waves with the same energy. We will look at the geometric properties of these random functions such as topology and geometry of nodal domains and nodal lines. In particular we will show that nodal domains of any given topological structure or with any given area occur with positive density.