No smooth Phase Transition for the Nodal Length of Band-limited Spherical Random Fields

TL;DR

This paper studies the variance of the nodal length of band-limited spherical random fields, revealing a non-smooth transition in variance growth as the frequency window changes, with implications for understanding eigenfunction behavior.

Contribution

It demonstrates that the asymptotic variance remains logarithmic when the frequency window shrinks sublinearly, contradicting the expectation of a smooth transition.

Findings

Variance is linear with frequency for large eigenfunction sets.

Variance is logarithmic when considering a single eigenfunction.

No smooth transition in variance growth as the frequency window varies.

Abstract

In this paper, we investigate the variance of the nodal length for band-limited spherical random waves. When the frequency window includes a number of eigenfunctions that grows linearly, the variance of the nodal length is linear with respect to the frequency, while it is logarithmic when a single eigenfunction is considered. Then, it is natural to conjecture that there exists a smooth transition with respect to the number of eigenfunctions in the frequency window; however, we show here that the asymptotic variance is logarithmic whenever this number grows sublinearly, so that the window "shrinks". The result is achieved by exploiting the Christoffel-Darboux formula to establish the covariance function of the field and its first and second derivatives. This allows us to compute the two-point correlation function at high frequency and then to derive the asymptotic behaviour of the variance.