High-dimensional Monochromatic Random Waves approximate the Bargmann-Fock field

Fix d ∈ N and let n ∈ N be a large integer. Let f d,n be the restriction to R d × {0} n−d of the Monochromatic Random Wave, i.e. the unique a.s. smooth, centered stationary Gaussian field whose spectral measure is the uniform measure on the unit sphere in R n. Then, we show that as n → +∞, f d,n approximates a Gaussian field g d,n whose spectral measure is a Gaussian measure with adequately chosen variance. As an application of this result, we deduce that for − > 0 small enough and n ∈ N large enough, the excursion sets {f d,n + > 0} percolate with probability one