Abstract:We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains.
“…The proof of this lemma is essentially the same as that of the so-called 'integral-geometric sandwich' (see [10,Lemma 1]). Since we are using boxes rather than balls, we give a direct proof for the reader's convenience.…”
Section: Proof Of Lemma 31: Most Components Arise Locallymentioning
confidence: 99%
“…where c 0 > 0 is some positive constant borrowed from theory of random fields ("the universal Nazarov-Sodin constant" [10], see Sect. 1.3 below).…”
Section: Nodal Domainsmentioning
confidence: 99%
“…(Equivalently, μ is the spectral measure of h μ .) Now for R > 0 let N hμ (R) be the number of nodal domains of h μ lying entirely inside the disc B(R) centred at the origin of radius R. Nazarov and Sodin [10,Theorem 1] proved that there exists a non-negative number c μ = c NS (μ) ≥ 0 such that as R → ∞ the expected number of nodal domains lying in B(R) satisfies the asymptotic E[N hμ (R)] = c μ · πR 2 (1 + o(1)), (1.9) and gave some very mild conditions on μ that guarantee that c μ is strictly positive; they also proved the almost sure convergence and convergence in mean of N hμ (R) πR 2 to c μ , under some extra regularity assumptions on μ. The constant c 0 mentioned in Sect.…”
Section: The Nazarov-sodin Constantmentioning
confidence: 99%
“…1.2. Nazarov and Sodin [10] showed that it is quite easy to check whether for a given symmetric measure μ ∈ P(S 1 ) the corresponding NazarovSodin constant c NS (μ) = 0, and hence whether for a given set K ⊆ P(S 1 ) the image c NS (K) is bounded away from zero. …”
Section: Definition 12 (Constraint On the Coefficients)mentioning
confidence: 99%
“…Our final ingredient is an application of the work of Nazarov and Sodin [10], who have comprehensively studied nodal domains of Gaussian fields. We do this in two steps, the details of which are given in Sect.…”
We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.
“…The proof of this lemma is essentially the same as that of the so-called 'integral-geometric sandwich' (see [10,Lemma 1]). Since we are using boxes rather than balls, we give a direct proof for the reader's convenience.…”
Section: Proof Of Lemma 31: Most Components Arise Locallymentioning
confidence: 99%
“…where c 0 > 0 is some positive constant borrowed from theory of random fields ("the universal Nazarov-Sodin constant" [10], see Sect. 1.3 below).…”
Section: Nodal Domainsmentioning
confidence: 99%
“…(Equivalently, μ is the spectral measure of h μ .) Now for R > 0 let N hμ (R) be the number of nodal domains of h μ lying entirely inside the disc B(R) centred at the origin of radius R. Nazarov and Sodin [10,Theorem 1] proved that there exists a non-negative number c μ = c NS (μ) ≥ 0 such that as R → ∞ the expected number of nodal domains lying in B(R) satisfies the asymptotic E[N hμ (R)] = c μ · πR 2 (1 + o(1)), (1.9) and gave some very mild conditions on μ that guarantee that c μ is strictly positive; they also proved the almost sure convergence and convergence in mean of N hμ (R) πR 2 to c μ , under some extra regularity assumptions on μ. The constant c 0 mentioned in Sect.…”
Section: The Nazarov-sodin Constantmentioning
confidence: 99%
“…1.2. Nazarov and Sodin [10] showed that it is quite easy to check whether for a given symmetric measure μ ∈ P(S 1 ) the corresponding NazarovSodin constant c NS (μ) = 0, and hence whether for a given set K ⊆ P(S 1 ) the image c NS (K) is bounded away from zero. …”
Section: Definition 12 (Constraint On the Coefficients)mentioning
confidence: 99%
“…Our final ingredient is an application of the work of Nazarov and Sodin [10], who have comprehensively studied nodal domains of Gaussian fields. We do this in two steps, the details of which are given in Sect.…”
We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.