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Correlation Structure and Resonant Pairs for Arithmetic Random Waves. (arXiv:2312.13166v1 [math.PR])

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The geometry of Arithmetic Random Waves has been extensively investigated in
the last fifteen years, starting from the seminal papers [RW08, ORW08]. In this
paper we study the correlation structure among different functionals such as
nodal length, boundary length of excursion sets, and the number of intersection
of nodal sets with deterministic curves in different classes; the amount of
correlation depends in a subtle fashion from the values of the thresholds
considered and the symmetry properties of the deterministic curves. In
particular, we prove the existence of resonant pairs of threshold values where
the asymptotic correlation is full, that is, at such values one functional can
be perfectly predicted from the other in the high energy limit. We focus mainly
on the 2-dimensional case but we discuss some specific extensions to dimension
3.

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