arXiv subjects

Igor Wigman

Publications and source records attributed to Igor Wigman.

59 records · Page 4Linked to original sources

arXiv2007

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 $4π^2\eigenvalue$ 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.

math-ph↗

arXiv2007

The Leray measure of nodal sets for random eigenfunctions on the torus

We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in 2 or more dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with multiplicity N tending to infinity. The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant. Our main result is that the variance of Leray measure is asymptotically 1/(4 pi N), as N tends to infinity, at least in dimensions 2 and at least 5.

math-ph↗

arXiv2006

Statistics of lattice points in thin annuli for generic lattices

We study the statistical properties of the counting function of lattice points inside thin annuli. By a conjecture of Bleher and Lebowitz, if the width shrinks to zero, but the area converges to infinity, the distribution converges to the Gaussian distribution. If the width shrinks slowly to zero, the conjecture was proven by Hughes and Rudnick for the standard lattice, and in our previous paper for generic rectangular lattices. We prove this conjecture for arbitrary lattices satisfying some generic Diophantine properties, again assuming the width of the annuli shrinks slowly to zero. One of the obstacles of applying the technique of Hughes-Rudnick on this problem is the existence of so-called close pairs of lattice points. In order to overcome this difficulty, we bound the rate of occurence of this phenomenon by extending some of the work of Eskin-Margulis-Mozes on the quantitative Openheim conjecture.

math.NT↗

arXiv2005

The distribution of lattice points in elliptic annuli

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

math.NT↗

arXiv2003

Counting Singular Matrices with Primitive Row Vectors

We solve an asymptotic problem in the geometry of numbers, where we count the number of singular $n\times n$ matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as $T \to \infty $, the number is asymptotic to $ \frac{(n-1)u_n}{ζ(n) ζ(n-1)^{n}}T^{n^{2}-n}\log (T)$ for $n \ge 3$. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. Schmidt.

math.NT↗