Spectral quasi correlations and phase-transitions for the nodal length of Arithmetic Random Waves

Abstract: Spectral quasi correlations are small sums of lattice points lying on the same circle; we show that, for generic integers representable as the sum of two squares, there are no spectral quasi-correlations. Moreover, we apply our result to study the nodal length of Arithmetic Random Waves at small scales: we show that there exists a phase-transition for the distribution of the nodal length at a logarithmic power above Planck-scale. Furthermore, we give strong evidence for the existence of an intermediate phase between Arithmetic and Berry's random waves.