The distribution of lattice points in elliptic annuli

Let $N(t, \rho)$ be the number of lattice points in a thin elliptical annuli. We assume the aspect ratio $\beta$ of the ellipse is transcendental and Diophantine in a strong sense (this holds for {\em almost all} aspect ratios). The variance of $N(t, \rho)$ is $t(8\pi \beta \cdot \rho)$. We show that if $\rho$ shrinks slowly to zero then the distribution of the normalized counting function $\frac{N(t, \rho) - A(2t\rho+\rho^2)}{\sqrt{8 \pi \beta \cdot t \rho}}$ is Gaussian, where A is the area of the ellipse. The case of \underline{circular} annuli is due to Hughes and Rudnick

On the expected Betti numbers of the nodal set of random fields

This note concerns the asymptotics of the expected total Betti numbers of the nodal set for an important class of Gaussian ensembles of random fields on Riemannian manifolds. By working with the limit random field defined on the Euclidean space we were able to obtain a locally precise asymptotic result, though due to the possible positive contribution of large percolating components this does not allow to infer a global result. As a by-product of our analysis, we refine the lower bound of Gayet-Welschinger for the important Kostlan ensemble of random polynomials and its generalisation to K\"{a}hler manifolds.Comment: 18 pages, 1 figur

Fluctuations of the nodal length of random spherical harmonics, erratum

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.Comment: This is to correct a sign mistake that has been made in the previous version (that was published in Comm. Math. Phys.). As a result the leading constant in all the theorems was wrong, and the constants are now consistent with the one predicted by Berry. A corrected manuscript plus a detailed erratum with all the corrections that were made relatively to the version published is attache

On the volume of nodal sets for eigenfunctions of the Laplacian on the torus

We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues 4\pi^2\eigenvalue with growing multiplicity \Ndim\to\infty, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is const \sqrt{\eigenvalue}. Our main result is that the variance of the volume normalized by \sqrt{\eigenvalue} is bounded by O(1/\sqrt{\Ndim}), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.Comment: 20 pages, Was accepted for publication in the Annales Henri Poincar

The distribution of the zeroes of random trigonometric polynomials

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that \frac{Z_{N}-\E Z_{N}}{\sqrt{cN}} converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.Comment: 51 pages. We cut the size of the paper to better suit publication. In particular, all the results of empirical experiments were cut off. Some standard results in probability and stochastic processes were also omitted. Numerous typos and mistakes were corrected following the suggestions of referees. This paper was accepted for publication in the American Journal of Mathematics

Topologies of nodal sets of random band limited functions

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.Comment: An announcement of recent results. Includes an announcement of the resolution of some open questions from the older version. 11 pages, 6 figure

Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves

This is a manuscript containing the full proofs of results announced in [KW], together with some recent updates. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions.Comment: 27 pages, 6 figures. To appear in Advances in Mathematic