Maurizia Rossi

Bio: Maurizia Rossi is an academic researcher from University of Luxembourg. The author has contributed to research in topics: Central limit theorem & Spherical harmonics. The author has an hindex of 12, co-authored 40 publications receiving 496 citations. Previous affiliations of Maurizia Rossi include Paris Descartes University & University of Pisa.

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

Abstract: “Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincare 9(1):109–130, 2008; Krishnapur et al. in Annals of Mathematics (2) 177(2):699–737, 2013). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener–Ito 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). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics

Abstract: Nous etudions le comportement asymptotique de la longueur nodale de fonctions propres aleatoires $f_{\ell}$ du Laplacien spherique pour valeurs propres tres eleves $\ell\rightarrow+\infty$, c’est-a-dire la longueur de leur ensemble de niveau zero $f_{\ell}^{-1}(0)$. Nous demontrons que la longueur nodale est asymptotiquement equivalente, au sens de $L^{2}$, au « sample trispectrum », c’est-a-dire l’integral de $H_{4}(f_{\ell}(x))$, le polynome de Hermite d’ordre quatre evalue en $f_{\ell}$. Une consequence de ce resultat est un Theoreme Central Limite quantitatif (dans le sens de la distance de Wasserstein) pour la longueur nodale, quand l’energie tend vers l’infini.

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

Abstract: "Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013)). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener-Ito chaos expansion for the nodal length shows that it is dominated by its $4$th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

Nodal Statistics of Planar Random Waves

Abstract: We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $$E > 0$$ , both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ( $$E \rightarrow \infty$$ ). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat 2-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by Canzani and Hanin (2016), in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002).

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

Abstract: “Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincare 9(1):109–130, 2008; Krishnapur et al. in Annals of Mathematics (2) 177(2):699–737, 2013). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener–Ito 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). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions

Abstract: We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler–Poincare characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler–Poincare characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level $u$ is fully degenerate, that is, the Euler–Poincare characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz–Killing curvatures of the excursion sets for Gaussian spherical harmonics.

Fluctuations of the nodal length of random spherical harmonics

Abstract: Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree $n$ having Laplace eigenvalue $E=n(n+1)$. We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order $n$. It is natural to conjecture that the variance should be of order $n$, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order $\log{n}$. This behaviour is consistent with the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In addition we find that a similar result is applicable for "generic" linear statistics of the nodal lines.