arXiv subjects
Publications and source records attributed to Maurizia Rossi.
We study the correlation between the nodal length of random spherical harmonics and the measure of the boundary for excursion sets at any non-zero level. We show that the correlation is asymptotically zero, while the partial correlation after controlling for the random $L^2$-norm on the sphere of the eigenfunctions is asymptotically one.
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-sphere ($d\ge 2$). We investigate the distribution of their defect i.e., the difference between the measure of positive and negative regions. Marinucci and Wigman studied the two-dimensional case giving the asymptotic variance (Marinucci and Wigman 2011) and a Central Limit Theorem (Marinucci and Wigman 2014), both in the high-energy limit. Our main results concern asymptotics for the defect variance and quantitative CLTs in Wasserstein distance, in any dimension. The proofs are based on Wiener-Itô chaos expansions for the defect, a careful use of asymptotic results for all order moments of Gegenbauer polynomials and Stein-Malliavin approximation techniques by Nourdin and Peccati. Our argument requires some novel technical results of independent interest that involve integrals of the product of three hyperspherical harmonics.
We study the nodal intersections number of random Gaussian toral Laplace eigenfunctions ("arithmetic random waves") against a fixed smooth reference curve. The expected intersection number is proportional to the the square root of the eigenvalue times the length of curve, independent of its geometry. The asymptotic behaviour of the variance was addressed by Rudnick-Wigman; they found a precise asymptotic law for "generic" curves with nowhere vanishing curvature, depending on both its geometry and the angular distribution of lattice points lying on circles corresponding to the Laplace eigenvalue. They also discovered that there exist peculiar "static" curves, with variance of smaller order of magnitude, though did not prescribe what the true asymptotic behaviour is in this case. In this paper we study the finer aspects of the limit distribution of the nodal intersections number. For "generic" curves we prove the Central Limit Theorem (at least, for "most" of the energies). For the aforementioned static curves we establish a non-Gaussian limit theorem for the distribution of nodal intersections, and on the way find the true asymptotic behaviour of their fluctuations, under the well-separatedness assumption on the corresponding lattice points, satisfied by most of the eigenvalues.
In this survey we collect some of the recent results on the "nodal geometry" of random eigenfunctions on Riemannian surfaces. We focus on the asymptotic behavior, for high energy levels, of the nodal length of Gaussian Laplace eigenfunctions on the torus (arithmetic random waves) and on the sphere (random spherical harmonics). We give some insight on both Berry's cancellation phenomenon and the nature of nodal length second order fluctuations (non-Gaussian on the torus and Gaussian on the sphere) in terms of chaotic components. Finally we consider the general case of monochromatic random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian surface with frequencies from a short interval, whose scaling limit is Berry's Random Wave Model. For the latter we present some recent results on the asymptotic distribution of its nodal length in the high energy limit (equivalently, for growing domains).
We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell}$ of high degree $\ell \rightarrow\infty$, i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It is found that the nodal lengths are asymptotically equivalent, in the $L^{2}$-sense, to the "sample trispectrum", i.e., the integral of $H_{4}(f_{\ell}(x))$, the fourth-order Hermite polynomial of the values of $f_{\ell}$. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.
We consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus (arithmetic random waves), and provide explicit Berry-Esseen bounds in the 1-Wasserstein distance for the normal and non-normal high-energy approximation of the associated Leray measures and total nodal lengths, respectively. Our results provide substantial extensions (as well as alternative proofs) of findings by Oravecz, Rudnick and Wigman (2007), Krishnapur, Kurlberg and Wigman (2013), and Marinucci, Peccati, Rossi and Wigman (2016). Our techniques involve Wiener-Ito chaos expansions, integration by parts, as well as some novel estimates on residual terms arising in the chaotic decomposition of geometric quantities that can implicitly be expressed in terms of the coarea formula.
"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-Itô 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.
In this PhD Thesis we investigate the geometry of random fields on compact Riemannian manifolds, in particular the two-dimensional sphere. In the first part, we characterize isotropic Gaussian fields on homogeneous spaces of a compact group and then we prove the non-existence of P. Lévy's Brownian field on the group SO(3), which moreover allows to extend the same kind of result to SO(n), SU(n) for n bigger than 3. In the second part, we investigate the high-energy behavior of random eigenfunctions on the (hyper)sphere and on the torus proving quantitative CLT results for some geometric functionals of those eiegenfunctions. A nice non-Central and non-Universal result is shown for nodal length distribution in the case of arithmetic random waves. Finally, we extend representation formulas obtained in the first part to the case of spin random fields on the sphere, introducing a new approach i.e., the pullback random field, that easily allows to study random sections of homogeneous vector bundles.
We provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a $d$-dimensional general diffusion process $X$, as the conditioning time tends to $0$. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift $b$ of $X$. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided $b$ enjoyes a simple condition that is always satisfied in dimension $1$. On the other hand, we show that the drift can be influential if this assumption is not satisfied. }
The Berry heuristic has been a long standing \emph{ansatz} about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see Berry 1977). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace-Beltrami operator on the normalized $d$-dimensional sphere - also known as spherical harmonics). Some applications to non-Gaussian models are also discussed.
In this short survey we recollect some of the recent results on the high energy behavior (i.e., for diverging sequences of eigenvalues) of nonlinear functionals of Gaussian eigenfunctions on the $d$-dimensional sphere $\mathbb S^d$, $d\ge 2$. We present a quantitative Central Limit Theorem for a class of functionals whose Hermite rank is two, which includes in particular the empirical measure of excursion sets in the non-nodal case. Concerning the nodal case, we recall a CLT result for the defect on $\mathbb S^2$. The key tools are both, the asymptotic analysis of moments of all order for Gegenbauer polynomials, and so-called Fourth-Moment theorems.
We provide Large Deviation estimates for the bridge of a $d$-dimensional general diffusion process as the conditioning time tends to $0$ and apply these results to the evaluation of the asymptotics of its exit time probabilities. We are motivated by applications to numerical simulation, especially in connection with stochastic volatility models.
We investigate Stein-Malliavin approximations for nonlinear functionals of geometric interest of Gaussian random eigenfunctions on the unit $d$ -dimensional sphere ${\mathbb{S}}^{d},$ $d\geq 2.$ All our results are established in the high energy limit, i.e. for eigenfunctions corresponding to growing eigenvalues. More precisely, we provide an asymptotic analysis for the variance of random eigenfunctions, and also establish rates of convergence for various probability metrics for Hermite subordinated processes, arbitrary polynomials of finite order and square integral nonlinear transforms; the latter, for instance, allows to prove a quantitative Central Limit Theorem for the excursion area. Some related issues were already considered in the literature for the $2$-dimensional case ${\mathbb{S}}^{2}$; our results are new or improve the existing bounds even for this special case. Proofs are based on the asymptotic analysis of moments of all order for Gegenbauer polynomials, and make extensive use of the recent literature on so-called fourth-moment theorems by Nourdin and Peccati.
We develop a technique for the construction of random fields on algebraic structures. We deal with two general situations: random fields on homogeneous spaces of a compact group and in the spin-line bundles of the 2-sphere. In particular, every spin Gaussian isotropic field can be obtained with this construction. Our construction extends P. Lévy's original idea for the spherical Brownian Motion.
We investigate positive definiteness of the Brownian kernel K(x,y)=1/2(d(x,x_0) + d(y,x_0) - d(x,y)) on a compact group G and in particular for G=SO(n).