We consider Random Wave models, that is probability distributions on Laplacian eigenfunctions, on homogeneous spaces such as Euclidean spaces, Hyperspheres and Hyperbolic spaces. Determining the asymptotic behavior at large frequency of functionals of these objects is complicated by the oscillatory nature of their covariance functions. Oscillatory integrals appearing in evaluating variances of said functionals turn out to be closely related to integral representations of densities of uniform random walks in Euclidean spaces, and this connection can be exploited to deduce results on fluctuations of integral functionals of Random Waves.

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