Spectral Quasi Correlations and Phase Transitions for the Nodal Length of Arithmetic Random Waves

Abstract

Abstract We study the nodal length of arithmetic random waves at small scales: we show that there exists a phasetransition 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. These results are based on the study of small sums of lattice points lying on the same circle, called spectral quasi correlations. We show that, for generic integers representable as the sum of two squares, there are no spectral quasi correlations.