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# Fluctuations of Polyspectra in Spherical and Euclidean Random Wave Models

Mar 2023

We consider polynomial transforms (polyspectra) of Berry's model -- theEuclidean Random Wave model -- and of Random Hyperspherical Harmonics. Wedetermine the asymptotic behavior of variance for polyspectra of any order inthe high-frequency limit. In particular, we are able to treat polyspectra ofany odd order $q\geq 5$, whose asymptotic behavior was left as a conjecture inthe case of Random Hyperspherical Harmonics by Marinucci and Wigman(\emph{Comm. Math. Phys.} 2014). To this end, we exploit a relation between thevariance of polyspectra and the distribution of uniform random walks onEuclidean space with finitely many steps, which allows us to rely on technicalresults in the latter context.

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