Counting nodal domains

TL;DR

This paper introduces a method to analyze the distribution and connectivity of nodal domains in 2D Gaussian random waves, with implications for understanding their universality class and correlation properties.

Contribution

It presents a novel analytical approach using auxiliary Potts-spins to study nodal domain distributions and connectivity in Gaussian random waves.

Findings

Calculated distribution of nodal domains

Identified weak long-range correlations

Proposed link to percolation universality class

Abstract

We consider the nodal domains of Gaussian random waves in two dimensions. We present a method to calculate the distribution of the number of nodal domains and the average connectivity with the help of auxiliary Potts-spins. An analytical approach could be helpful to decide whether the pattern of nodal domains belongs to the universality class of short-ranged percolation. This is not completely evident since we find (weak) long-ranged correlations between distant avoided nodal intersections.