Seminar on Probability and Statistics

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Organizer(s) Nakahiro Yoshida, Teppei Ogihara, Yuta Koike

2016/11/01

11:30-12:30   Room #123 (Graduate School of Math. Sci. Bldg.)
Giovanni Peccati (University du Luxembourg)
Second order fluctuations for zeros of arithmetic random waves
[ Abstract ]
Originally introduced by Rudnick and Wigman (2007), arithmetic random waves are Gaussian Laplace eigenfunctions on the two-dimensional torus. In this talk, I will describe the high-energy behaviour of the so-called « nodal length » (that, is the volume of the zero set) of such random objects, and show that (quite unexpectedly) it is non-central and non-universal. I will also discuss the connected problem of counting the number of intersections points of independent nodal sets (equivalent to « phase singularities » for complex waves) in the high-energy regime. Both issues are tightly connected to the arithmetic study of lattice points on circles. One key concept in our presentation is that of ‘Berry cancellation phenomenon’ (see M.V. Berry, 2002), for which an explanation in terms of chaos expansions and integration by parts (Green formula) will be provided. Based on joint works (GAFA 2016 & Preprint 2016) with D. Marinucci (Rome Tor Vergata), M. Rossi (Luxembourg) and I. Wigman (King’s College, London), and with F. Dalmao (University of Uruguay), I. Nourdin (Luxembourg) and M. Rossi (Luxembourg).