In recent years, considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In this paper, we extend those results to proper subsets of the sphere S2, i.e., spherical caps, focussing, in particular, on the excursion area. Precisely, we show that the asymptotic behaviour of the excursion area is dominated by the so-called second-order chaos component and we exploit this result to establish a quantitative central limit theorem, in the high energy limit. These results generalize analogous findings for the full sphere; their proofs, however, require more sophisticated techniques, in particular, a careful analysis (of some independent interest) for smooth approximations of the indicator function for spherical cap subsets.

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We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in and Todino (2020), respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thäle (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence

We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020+) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thäle (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.

A lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high‐frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz‐Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler‐Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so‐called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data.