Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

par Maurizia Rossi (Université du Luxembourg)

mercredi 5 avr. 2017, 14:00 → 15:00 Europe/Paris

Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two- dimensional torus. We are interested in the distribution of the length of their nodal lines. In [1] the authors prove that the asymptotics for the variance is non-universal. Their result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy. In this talk we show that the nodal length converges to a non-universal (non- Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles [2]. Our argument has two main ingredients. An explicit derivation of the Wiener-Itô chaos expansion for the nodal length shows that it is dominated by its 4th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes - this is closely related to the so-called “obscure” Berry’s cancellation phenomenon). The rest of the argument relies on the precise analysis of the fourth order chaotic component. If time permits, we will discuss also some new findings of [3], where an alternative proof for the asymptotic variance in [1] is given, and rates of convergence for limit theorems in [2] are proved. This talk is mainly based on joint works with Domenico Marinucci (Università di Roma Tor Vergata), Giovanni Peccati (Université du Luxembourg) and Igor Wigman (King’s College London). ------------------------------------------------------------ References ------------------------------------------------------------ [1] M. Krishnapur, P. Kurlberg, I. Wigman, Nodal length fluctuations for arithmetic random waves, Annals of Mathematics (2) 177 (2013), no. 2, 699–737. ------------------------------------------------------------ [2] D. Marinucci, G. Peccati, M. Rossi, I. Wigman, Non-universality of nodal length distribution for arithmetic random waves, Geometric and Functional Analysis 26 (2016), no. 3, 926–960. ------------------------------------------------------------ [3] G. Peccati, M. Rossi, Quantitative limit theorems for local functionals of arithmetic random waves, Preprint arXiv: 1702.03765, (2017).