Fluctuations of polyspectra in spherical and Euclidean random wave models

We consider polynomial transforms (polyspectra) of Berry’s model – the Euclidean Random Wave model – and of Random Hyperspherical Harmonics. We determine the asymptotic behavior of variance for polyspectra of any order in the high-frequency limit. In particular, we are able to treat polyspectra of any odd order q ≥ 5, whose asymptotic behavior was left as a conjecture in the case of Random Hyperspherical Harmonics by Marinucci and Wigman (Comm. Math. Phys. 2014). To this end, we exploit a relation between the variance of polyspectra and the distribution of uniform random walks on Euclidean space with finitely many steps, which allows us to rely on technical results in the latter context.