Planck-scale distribution of nodal length of arithmetic random waves

TL;DR

This paper investigates the behavior of the nodal length of arithmetic random waves at Planck scale, revealing asymptotic laws and correlations that enable partial reconstruction of total nodal length, with novel number-theoretic methods.

Contribution

It introduces a detailed analysis of nodal length at Planck scale, showing asymptotic correlation with total length and employing spectral Quasi-Correlations for the first time in this context.

Findings

Variance of restricted nodal length matches total nodal length asymptotics.

Restricted and total nodal lengths are asymptotically fully correlated.

Spectral Quasi-Correlations are used to derive key bounds.

Abstract

We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic" energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use of bounds for the so-called spectral Quasi-Correlations, i.e. unusually small sums of lattice points lying on the same circle.