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The Leray measure of nodal sets for random eigenfunctions on the torus. (English) Zbl 1153.35058
For $$E>0$$, let $$N(E)$$ be the dimension of the eigenspace of the Laplacian on the $$d$$-dimensional torus, i.e. $$N(E)$$ is the number of the set of all $$\lambda\in \mathbb Z^d$$ such that $$| \lambda| ^2 =E.$$ Let $$b_\lambda, c_\lambda$$ be independent standard Gaussian random variables such that $$b_{-\lambda}=b_\lambda$$ and $$c_{-\lambda} =-c_\lambda$$, and let $$L(f)$$ be the Leray measure of the nodal set of the random eigenfunction
$f(x):= (2N(E))^{-1/2} \sum_{\lambda\in \mathbb Z^d: | \lambda | ^2=E} b_\lambda \cos 2\pi I \lambda\cdot x -c_\lambda \sin 2 \pi i \lambda\cdot x.$ The introduced Leray measure has the property that $$\mathbb E L(f)= (2\pi)^{-1/2}$$ and $$\text{Var}(L(f))$$ behaves as $$(4\pi N(E))^{-1}$$ as $$N\to \infty$$.

##### MSC:
 35P20 Asymptotic distributions of eigenvalues in context of PDEs 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
##### Keywords:
Leray measure; nodal set; random eigenfunction; torus
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