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We investigate the nodal volume of random hyperspherical harmonics $\lbraceT_{\ell;d}\rbrace_{\ell\in \mathbb N}$ on the $d$-dimensional unit sphere($d\ge 2$). We exploit an orthogonal expansion in terms of Laguerrepolynomials; this representation entails a drastic reduction in thecomputational complexity and allows to prove \emph{isotropy} for chaoticcomponents, an issue which was left open in the previous literature. As afurther application, we establish our main result, i.e., variance bounds forthe 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 theso-called Berry's cancellation phenomenon holds in any dimension: namely, thenodal variance is one order of magnitude smaller than the variance of thevolume of level sets at any non-zero threshold, in the high-energy limit.

Laguerre Expansion for Nodal Volumes and Applications