2,718 research outputs found
Asymptotic distribution of nodal intersections for arithmetic random waves
Full text linkWe 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
Fluctuations of the Total Number of Critical Points of Random Spherical Harmonics
Full text linkWe determine the asymptotic law for the fluctuations of the total number of
critical points of random Gaussian spherical harmonics in the high degree
limit. Our results have implications on the sophistication degree of an
appropriate percolation process for modelling nodal domains of eigenfunctions
on generic compact surfaces or billiards
On the Distribution of the Critical Values of Random Spherical Harmonics
Full text linkWe study the limiting distribution of critical points and extrema of random spherical harmonics, in the high energy
limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions
for the variances and we show that the empirical measures in the high-energy limits converge weakly to their expected
values. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances,
entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed
On the area of excursion sets of spherical Gaussian eigenfunctions
No full textThe high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivation arising from physics and cosmology. In this paper, we are concerned with the high frequency behaviour of excursion sets; in particular, we establish a uniform central limit theorem for the empirical measure, i.e., the proportion of spherical surface, where spherical Gaussian eigenfunctions lie below a level z. Our proofs borrow some techniques from the literature on stationary long memory processes; in particular, we expand the empirical measure into Hermite polynomials, and establish a uniform weak reduction principle, entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion. As a result, we establish a functional central limit theorem; the limiting process is fully degenerate.
© 2011 American Institute of Physic
Two Point Function for Critical Points of a Random Plane Wave
Full text linkRandom plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemannian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius
Nodal deficiency of random spherical harmonics in presence of boundary
Full text link=We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere, satisfying the Dirichlet boundary conditions along the equator. For this model, we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for the total nodal length of this ensemble relative to the rotation invariant model of random spherical harmonics. Jean Bourgain’s research, and his enthusiastic approach to the nodal geometry of Laplace eigenfunctions, has made a crucial impact in the field and the current trends within. His works on the spectral correlations {Theorem 2.2 in the work of Krishnapur et al. [Ann. Math. 177(2), 699–737 (2013)]} and Bombieri and Bourgain [Int. Math. Res. Not. (IMRN) 11, 3343–3407 (2015)] have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavor, such as the torus or the square. Furthermore, Bourgain’s work [J. Bourgain, Isr. J. Math. 201(2), 611–630 (2014)] on toral Laplace eigenfunctions, also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts
No repulsion between critical points for planar Gaussian random fields
Full text linkWe study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a ‘generic’ field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index
The defect variance of random spherical harmonics
No full textThe defect of a function [IMAGE] is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is evaluating the defect variance, asymptotically in the high-frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern cosmological data analysis
Non-Universality of Nodal Length Distribution for Arithmetic Random Waves
Full text link"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-It\^o 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
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