On the number of nodal domains of random spherical harmonics

By Fedor Nazarov, M. Sodin. 2007

226 citations recorded by the source corpus.

Let $N(f)$ be a number of nodal domains of a random Gaussian spherical harmonic $f$ of degree $n$. We prove that as $n$ grows to infinity, the mean of $N(f)/n^2$ tends to a positive constant $a$, and that $N(f)/n^2$ exponentially concentrates around $a$. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal domains of random Gaussian plane waves.

Subfields: Atmospheric Science, Earth-Surface Processes, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/58d116c7d50ee89b1e86555d28a077b1bff562cf

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In Atmospheric Science

In Earth-Surface Processes

In Geometry and Topology

Citations in the corpus

On Pleijel's Nodal Domain Theorem (2013, methodology): “small. From a rigorous point of view, it is not unreasonable to expect further improvements related to the application of the Faber-Krahn 3 inequality for random models (the work of Nazarov and Sodin [9], while establishing a limiting distribution for N n when n → ∞ for random spherical harmonics (o”
EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS (2013, methodology): “As in [16], we then compute the probability of the presence of closed affine real algebraic submanifolds, inspired by an approach of Nazarov and Sodin [25]; see also [20].”
Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions (2015, methodology): “3 In [22], we showed that for the Gaussian ensemble of spherical harmonics of large degree L on the two-dimensional sphere, P {∣∣L−2N(f L )− υ ∣∣ > ε} < C(ε)e−c(ε) dimHL ,”
Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d (2014, unknown): “Recall now the notion of the excursion volume of T (cid”
Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution (2013, unknown): “, we obtain an upper bound for N(ϕj): N(ϕj) ≤ j= Vol(M) 4π λj(1 +o(1)). When M is the unit sphere S2 and ϕis a random spherical harmonics, then N(ϕ) ∼ cλϕ holds almost surely for some constant c>0 [NS]. However, for an arbitrary Riemannian surface, it is not even known whether one can always ﬁnd ”
Nodal Domains of Maass Forms I (2012, unknown): “rsion for NK(f) for K a subset of X like the windows in Figure 1 and the prediction is remarkably accurate for these four forms. Based on this and other experimentation in different settings ([BS07], [NS09],[Nas11],[Ko12]) it seems that this prediction is probably correct, with either the above cons”
A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions (2016, background): “Random spherical harmonics arise naturally from Fourier analysis of isotropic spherical random fields and in the investigation of quantum chaos, and they have hence drawn quite a lot of interest in the last few years (see, for instance, [8, 19, 22, 24, 29, 32, 33]); as discussed below, we believe th”
The complexity of spherical p-spin models - a second moment approach (2015, background): “See, for example, Nazarov and Sodin [35, 36] and references therein.”
EIGENFUNCTIONS AND NODAL SETS (2012, background): “ an eigenfunction whose nodal set has the topological type of the union of curves. Note that these spherical harmonics have relatively few nodal domains compared to the Pleijel bound. It is proved in [NS] that random spherical harmonics have aN2 nodal components for some (undetermined) a>0. Morse”
Percolation of random nodal lines (2016, unknown): “• What is the mean number of finite connected components per square? In [34] and [35], the authors proved the existence of a non explicit asymptotic equivalence of the number of connected components of the nodal sets.”