Real and Complex Analysis

On random fields

“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.

/pdf/non-universality-of-nodal-length-distribution-for-arithmetic-21cq0h0zd0.pdf

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

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.

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

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.

Fluctuations of the nodal length of random spherical harmonics

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.

/pdf/the-asymptotic-equivalence-of-the-sample-trispectrum-and-the-1osvgdgv5n.pdf

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

"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.

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).

/pdf/nodal-statistics-of-planar-random-waves-3v9hr1gpsy.pdf

Nodal Statistics of Planar Random Waves

https://orbilu.uni.lu/bitstream/10993/24087/1/dom_mau.pdf

Maurizia Rossi

Papers

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

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

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

Nodal Statistics of Planar Random Waves

Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d