Nodal length fluctuations for arithmetic random waves

Krishnapur, Manjunath; Kurlberg, Pär; Wigman, Igor

doi:10.4007/annals.2013.177.2.8

Cited by 110 publications

(218 citation statements)

References 33 publications

(115 reference statements)

Supporting

Mentioning

211

Contrasting

Order By: Relevance

“…A beautiful topic that we do not discuss here is the zero sets of random eigenfunctions (linear combinations with random coefficients). The interested reader can start wtih [77], [81], [18], [93], [51] for the introduction to random eigenfunctions.…”

Section: Two Examplesmentioning

confidence: 99%

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Logunov

Malinnikova

2018

Current Developments in Mathematics

View full text Add to dashboard Cite

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.

show abstract

Section: Two Examplesmentioning

confidence: 99%

Review of Yau’s conjecture on zero sets of Laplace eigenfunctions

Logunov

Malinnikova

2018

Current Developments in Mathematics

View full text Add to dashboard Cite

show abstract

“…A 2 := {(μ(4),μ(8)) : μ is attainable} (5) denote the projection of the set of attainable measures onto the first two non-trivial Fourier coefficients. The intersection of A 2 with the vertical strip {(x, y) : |x| ≤ 1/3} turns out to have a rather complicated fractal structure with infinitely many spikessee Fig.…”

Section: Letmentioning

confidence: 99%

On probability measures arising from lattice points on circles

Kurlberg

Wigman

2016

Math. Ann.

Self Cite

View full text Add to dashboard Cite

A circle, centered at the origin and with radius chosen so that it has nonempty intersection with the integer lattice Z 2 , gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a "fractal" structure. This complicated structure in some sense arises from prime powers-singularities do not occur for circles of radius √ n if n is square free.

show abstract

“…For this model, it is known [5] that various local properties of f n , e.g., the total length of the nodal line f −1 n (0), depend on the limiting angular distribution of {λ ∈ Z 2 : λ 2 = n}. More precisely, for n ∈ S let μ n = 1 r 2 (n)…”

Section: Statement Of Results For Arithmetic Random Wavesmentioning

confidence: 99%

“…For μ ∈ P Symm , the maximal value d max is uniquely attained by c NS (μ S 1 ), where μ S 1 is the uniform measure on Motivated by the fact that the nodal length variance only depends on the first non-trivial Fourier coefficient of the measure [5], and some other local computations, we raise the following question.…”

Section: Conjecture 22mentioning

confidence: 99%