A relation between two different formulations of the Berry's conjecture

The Random Wave Conjecture of M. V. Berry is the heuristic that eigenfunctions of a classically chaotic system should behave like Gaussian random fields, in the large eigenvalue limit. In this work we collect some definitions and properties of Gaussian random fields, and show that the formulation of the Berry's conjecture proposed using local weak limits is equivalent to the one that is based on the Benjamini-Schramm convergence. Finally, we see that both these formulations of the Berry's property imply another property known as inverse localization that relates high energy eigenfunctions and solutions to the Euclidean Helmholtz equation.