# Functional Convergence of Berry's Nodal Lengths: Approximate Tightness and Total Disorder

@inproceedings{Notarnicola2022FunctionalCO, title={Functional Convergence of Berry's Nodal Lengths: Approximate Tightness and Total Disorder}, author={Massimo Notarnicola and Giovanni Peccati and Anna Vidotto}, year={2022} }

We consider Berry’s random planar wave model (1977), and prove spatial functional limit theorems – in the high-energy limit – for discretized and truncated versions of the random ﬁeld obtained by restricting its nodal length to rectangular domains. Our analysis is crucially based on a detailed study of the projection of nodal lengths onto the so-called second Wiener chaos , whose high-energy ﬂuctuations are given by a Gaussian total disorder ﬁeld indexed by polygonal curves. Such an exact…

## One Citation

### A note on $3d$-monochromatic random waves and cancellation

- Mathematics
- 2022

: In this note we prove that the asymptotic variance of the nodal length of monochromatic random waves restricted to an increasing domain in R 3 is linear in the volume of the domain. Put together…

## References

SHOWING 1-10 OF 50 REFERENCES

### Gaussian Random Measures Generated by Berry’s Nodal Sets

- MathematicsJournal of Statistical Physics
- 2020

We consider vectors of random variables, obtained by restricting the length of the nodal set of Berry’s random wave model to a finite collection of (possibly overlapping) smooth compact subsets of…

### Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

- Physics
- 2015

Abstract“Arithmetic random waves” are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman in Annales de l’Insitute Henri Poincaré 9(1):109–130, 2008; Krishnapur et…

### Fluctuations of the Nodal Length of Random Spherical Harmonics

- Mathematics
- 2009

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…

### Two point function for critical points of a random plane wave

- Mathematics
- 2017

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the…

### Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature

- Physics
- 2002

For real (time-reversal symmetric) quantum billiards, the mean length L of nodal line is calculated for the nth mode (n>>1), with wavenumber k, using a Gaussian random wave model adapted locally to…

### Local Weak Limits of Laplace Eigenfunctions

- Mathematics
- 2017

In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry's…

### Phase singularities in complex arithmetic random waves

- MathematicsElectronic Journal of Probability
- 2019

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-It\^o chaotic expansions in order to derive…

### Gaussian processes, kinematic formulae and Poincaré’s limit

- Mathematics
- 2009

We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the…

### Eigenfunctions and Random Waves in the Benjamini-Schramm limit

- Mathematics
- 2018

We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue…

### On the area of excursion sets of spherical Gaussian eigenfunctions

- Mathematics
- 2011

The high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivation arising from physics and…