Small Scale CLTs for the Nodal Length of Monochromatic Waves

@article{Dierickx2022SmallSC,
  title={Small Scale CLTs for the Nodal Length of Monochromatic Waves},
  author={Gauthier Dierickx and Ivan Nourdin and Giovanni Peccati and Maurizia Rossi},
  journal={Communications in Mathematical Physics},
  year={2022}
}
We consider the nodal length $L(\lambda)$ of the restriction to a ball of radius $r_\lambda$ of a {\it Gaussian pullback monochromatic random wave} of parameter $\lambda>0$ associated with a Riemann surface $(\mathcal M,g)$ without conjugate points. Our main result is that, if $r_\lambda$ grows slower than $(\log \lambda)^{1/25}$, then (as $\lambda\to \infty$) the length $L(\lambda)$ verifies a Central Limit Theorem with the same scaling as Berry's random wave model -- as established in Nourdin… 
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9 Citations

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References

SHOWING 1-10 OF 71 REFERENCES

A note on the reduction principle for the nodal length of planar random waves

The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics

We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell}$ of high degree $\ell \rightarrow\infty$, i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It…

Equidistribution of Random Waves on Small Balls

In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small scale expectation…

Universality of the nodal length of bivariate random trigonometric polynomials

We consider random trigonometric polynomials of the form \[ f_n(x,y)=\sum_{1\le k,l \le n} a_{k,l} \cos(kx) \cos(ly), \] where the entries $(a_{k,l})_{k,l\ge 1}$ are i.i.d. random variables that are…

Nodal Statistics of Planar Random Waves

AbstractWe consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $$E > 0$$E>0, both in the real and complex case, and prove limit theorems for the nodal statistics…

Nodal area distribution for arithmetic random waves

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3= \mathbb{R}^3/ \mathbb{Z}^3$ ($3$-dimensional 'arithmetic random waves'). We prove…

A logarithmic improvement in the two-point Weyl law for manifolds without conjugate points

In this paper, we study the two-point Weyl Law for the Laplace-Beltrami operator on a smooth, compact Riemannian manifold $M$ with no conjugate points. That is, we find the asymptotic behavior of the…

Small Scale Equidistribution of Random Eigenbases

We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e., eigenbases) on a compact manifold $${{\mathbb M}}$$M. Assume that the group of isometries acts…

Nodal lengths in shrinking domains for random eigenfunctions on $S^{2}$

We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: i.e., the length of the zero set…

Riemannian manifolds with maximal eigenfunction growth

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where…
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