Laguerre Expansion for Nodal Volumes and Applications

Todino Anna Paola, Università del Piemonte Orientale

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 isotropy for chaotic components, an issue which was left open in the previous literature. As a further application, we obtain an upper bound (that we conjecture to be sharp) for the asymptotic variance (as $\ell\to +\infty$) of the nodal volume for $d\geq3$. This result shows 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. Joint work with Domenico Marinucci and Maurizia Rossi.

Area: CS26 - Geometry of random fields (Michele Stecconi)

Keywords: Nodal sets; Laguerre polynomials; Asymptotic fluctuations; Berry’s cancellation phenomenon

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