We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue @math , 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 ( @math ). 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 @math -torus, see (2016) and (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).
Nodal length of random spherical harmonics . The Laplacian eigenvalues on the two-dimensional unit sphere are of the form @math , where @math , and the multiplicity of the @math -th eigenvalue is @math . The @math -th random eigenfunction (random spherical harmonic) on @math is a centered Gaussian field whose covariance kernel is @math where @math denotes the @math -th Legendre polynomial and @math the geodesic distance between the two points @math and @math (see @cite_11 ). The mean of the nodal length @math was computed in @cite_19 as @math while the asymptotic behaviour of the variance was derived in @cite_26 : as @math , @math The second order fluctuations of @math are Gaussian; more precisely, in @cite_0 it was shown that @math where @math is a standard Gaussian random variable.
@cite_0 @cite_19 @cite_26 @cite_11
2616288315 1808899688 2005820649 2040550846
We study the asymptotic behaviour of the nodal length of random @math -spherical harmonics @math of high degree @math , i.e. the length of their zero set @math . It is found that the nodal lengths are asymptotically equivalent, in the @math -sense, to the "sample trispectrum", i.e., the integral of @math , the fourth-order Hermite polynomial of the values of @math . A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit. 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. We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy @math in the localized phase. Assume the density of states function is not too flat near @math . Restrict it to some large cube @math . Consider now @math , a small energy interval centered at @math that asymptotically contains infintely many eigenvalues when the volume of the cube @math grows to infinity. We prove that, with probability one in the large volume limit, the eigenvalues of the random Hamiltonian restricted to the cube inside the interval are given by independent identically distributed random variables, up to an error of size an arbitrary power of the volume of the cube. As a consequence, we derive * uniform Poisson behavior of the locally unfolded eigenvalues, * a.s. Poisson behavior of the joint distibutions of the unfolded energies and unfolded localization centers in a large range of scales. * the distribution of the unfolded level spacings, locally and globally, * the distribution of the unfolded localization centers, locally and globally. We consider a discrete elliptic equation with random coefficients @math , which (to fix ideas) are identically distributed and independent from grid point to grid point @math . On scales large w. r. t. the grid size (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients @math are characterized by @math where the random field @math is the unique stationary solution of the ''corrector problem'' @math and @math denotes the ensemble average. It is known (''by ergodicity'') that the above ensemble average of the energy density @math , which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of @math on length scales @math is estimated as follows: @math where the averaging function (i. e. @math , @math ) has to be smooth in the sense that @math . In two space dimensions (i. e. @math ), there is a logarithmic correction. In other words, smooth averages of the energy density @math behave like as if @math would be independent from grid point to grid point (which it is not for @math ). This result is of practical significance, since it allows to estimate the error when numerically computing @math .