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

2020/05/10 by Andrea Sartori, Sartori, Andrea · 1 citation
Earth and Planetary Sciences · Mathematics · #Aeolian processes and effects #FOS: Mathematics #Geology and Paleoclimatology Research #Geometry and complex manifolds #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2005.04698

openalex publication_date 2020/05/10 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

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

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