Laguerre Expansion for Nodal Volumes and Applications

Abstract

We investigate the nodal volume of random hyperspherical harmonics $\lbrace T_{\ell;d}\rbrace_{\ell\in \mathbb N}$ on the $d$-dimensional unit sphere ($d\ge 2$). We exploit an orthogonal expansion in terms of Laguerre polynomials; this representation entails a drastic reduction in the computational complexity and allows to prove \emph{isotropy} for chaotic components, an issue which was left open in the previous literature. As a further application, we establish our main result, i.e., variance bounds for the nodal volume in any dimension; for $d\ge 3$ and as the eigenvalues diverge (i.e., as $\ell\to +\infty$), we obtain the upper bound $O(\ell^{-(d-2)})$ (that we conjecture to be \emph{sharp}). As a consequence, we show that the so-called Berry's cancellation phenomenon holds in any dimension: namely, the nodal variance is one order of magnitude smaller than the variance of the volume of level sets at any non-zero threshold, in the high-energy limit.