95 research outputs found
The distribution of lattice points in elliptic annuli
Let be the number of lattice points in a thin elliptical annuli.
We assume the aspect ratio of the ellipse is transcendental and
Diophantine in a strong sense (this holds for {\em almost all} aspect ratios).
The variance of is . We show that if
shrinks slowly to zero then the distribution of the normalized counting
function 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
having Laplace eigenvalue . We study the length distribution of the
nodal lines of random spherical harmonics. It is known that the expected length
is of order . It is natural to conjecture that the variance should be of
order , 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 . 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 of zeros of random
trigonometric polynomials of degree as . It is known that as
grows to infinity, the expected number of the zeros is asymptotic to
. The asymptotic form of the variance was predicted
by Bogomolny, Bohigas and Leboeuf to be for some . 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
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