Lipschitz-Killing Curvatures for Arithmetic Random Waves

TL;DR

This paper demonstrates that Lipschitz-Killing Curvatures of excursion sets for Arithmetic Random Waves are dominated by a single chaotic component at high frequencies, revealing full correlation and suggesting broader geometric functional formulas.

Contribution

It provides explicit formulas showing the dominance of a single chaotic component for Lipschitz-Killing Curvatures in high-frequency regimes of Arithmetic Random Waves.

Findings

Lipschitz-Killing Curvatures are dominated by a single chaotic component at high frequencies.

The dominant component is an explicit function of the threshold and the centered norm.

Geometric functionals are fully correlated in the high-energy limit.

Abstract

In this paper, we show that the Lipschitz-Killing Curvatures for the excursion sets of Arithmetic Random Waves (toral Gaussian eigenfunctions) are dominated, in the high-frequency regime, by a single chaotic component. The latter can be written as a simple explicit function of the threshold parameter times the centered norm of these random fields; as a consequence, these geometric functionals are fully correlated in the high-energy limit. The derived formulae show a clear analogy with related results on the round unit sphere and suggest the existence of a general formula for geometric functionals of random eigenfunctions on Riemannian manifolds.