Nodal length fluctuations for arithmetic random waves. (English) Zbl 1314.60101

The authors study the asymptotics of the variance of the nodal length of arithmetic random waves on the two dimensional torus \(\mathbb{T}\) (an arithmetic random wave is a random Gaussian field of real valued eigenfunctions \(f_{n}: \mathbb{T} \rightarrow \mathbb{R}\) corresponding to an eigenvalue of the Laplacian and its nodal length is defined as the length of \(f_{n}^{-1}(0)\)). The asymptotics are expressed in terms of the eigenvalue, of the dimension of the eigenspace, and of a constant related to a probability measure defined on the Laplace eigenspace. From the proved formulas one can deduce that the asymptotics of the variance are nonuniversal.


60G60 Random fields
58C40 Spectral theory; eigenvalue problems on manifolds
35P99 Spectral theory and eigenvalue problems for partial differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI arXiv


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