Volume distribution of nodal domains of random band-limited functions

TL;DR

This paper investigates how the volumes of nodal domains of random band-limited functions are distributed on manifolds, revealing a universal law in the high energy limit that is independent of the manifold's specifics.

Contribution

It establishes the existence of a universal volume distribution law for nodal domains of random band-limited functions on manifolds, with proven qualitative properties.

Findings

The volume distribution obeys a deterministic universal law at high energy.

The law's support, monotonicity, and continuity are rigorously established.

The distribution is independent of the underlying manifold in the high energy limit.

Abstract

We study the volume distribution of nodal domains of random band-limited functions on generic manifolds, and find that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.