No repulsion between critical points for planar Gaussian random fields

Beliaev, Dmitry; Cammarota, Valentina; Wigman, Igor

doi:10.1214/20-ecp362

Cited by 7 publications

(6 citation statements)

References 15 publications

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“…• In the particular case of the random plane wave (N = 2), our result coincides with the result of [11], formula (4) and [12], Theorem 1.2.…”

Section: Remarkssupporting

confidence: 87%

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Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices

Azais,

Delmas

2019

Preprint

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Let X = {X(t) : t ∈ R N } be an isotropic Gaussian random field with real values. In a first part we study the mean number of critical points of X with index k using random matrices tools. We obtain an exact expression for the probability density of the eigenvalue of rank k of a N -GOE matrix. We deduce some exact expressions for the mean number of critical points with a given index. In a second part we study attraction or repulsion between these critical points. A measure is the correlation function. We prove attraction between critical points when N > 2, neutrality for N = 2 and repulsion for N = 1. The attraction between critical points that occurs when the dimension is greater than two is due to critical points with adjacent indexes. A strong repulsion between maxima and minima is observed. The correlation function between maxima (or minima) depends on the dimension of the ambient space.

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“…• In the particular case of the random plane wave (N = 2), our result coincides with the result of [11], formula (4) and [12], Theorem 1.2.…”

Section: Remarkssupporting

confidence: 87%

“…In the particular case of the random plane wave (N = 2), the bound (44) is weaker than [11], formula (8), while (45) is sharper than [11], formula (5) or [12], Theorem 1.5.…”

Section: Correlation Function Between Critical Points With a Given Indexmentioning

confidence: 94%

Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices

Azais,

Delmas

2019

Preprint

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“…In the isotropic case, one may also entertain an intermediate notion of zero attraction or repulsion, in the situation when K 2 (0) > 0 is a finite, strictly positive number. For example, one may compare [9,10] this value at the origin of the 2-point correlation function corresponding to the critical points of F to that of the Poisson point process of the same intensity. Analogously to the above, one may introduce the k-point correlation function, k ≥ 3, to relate to the higher moments of the nodal volume.…”

Section: Main Ideas: Kac-rice Formulae and Their Applicationsmentioning

confidence: 99%

On the nodal structures of random fields -- a decade of results

Wigman¹

2022

Preprint

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“…Lemma 4.17. For s > 5 2 , there exists an explicit constant κ(s) > 0 such that Note that all the nonzero matrix components are exactly of order 1/r. While this fact does not make the problem any harder from a conceptual point of view, it leads to cumbersome expressions for the various quantities appearing in the equations.…”

Section: Note Thatmentioning

confidence: 99%

“…The asymptotic analysis of N (∇u, R) hinges on the celebrated Kac-Rice counting formula, which, under suitable technical hypotheses, expresses the expected number of zeros of a random field (in this case, the gradient ∇u) has in terms of a multivariate integral. As is well known, this formula has been used profusely in the literature [11,17,4,5], and in particular lies at the heart of the computation of EN (∇u, R) for s = 0 and of the finer asymptotics bounds for the expected number of extrema and saddle points and for higher order correlations obtained in [4] also in the translation-invariant case s = 0.…”

Section: Introductionmentioning

confidence: 99%

Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity

Enciso¹,

Peralta‐Salas²,

Romaniega³

2021

Preprint

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We consider Gaussian random monochromatic waves u on the plane depending on a real parameter s that is directly related to the regularity of its Fourier transform. Specifically, the Fourier transform of u is f dσ, where dσ is the Hausdorff measure on the unit circle and the density f is a function on the circle that, roughly speaking, has exactly s − 1 2 derivatives in L 2 almost surely. When s = 0, one recovers the classical setting for random waves with a translation-invariant covariance-kernel. The main thrust of this paper is to explore the connection between the regularity parameter s and the asymptotic behavior of the number N (∇u, R) of critical points that are contained in the disk of radius R 1. More precisely, we show that the expectation EN (∇u, R) grows like the area of the disk when the regularity is low enough (s < 32 ) and like the diameter when the regularity is high enough (s > 52 ), and that the corresponding exponent changes according to a linear interpolation law in the intermediate regime. The transitions occurring at the endpoint cases involve the square root of the logarithm of the radius. Interestingly, the highest asymptotic growth rate occurs only in the classical translation-invariant setting, s = 0. A key step of the proof of this result is the obtention of precise asymptotic expansions for certain Neumann series of Bessel functions. When the regularity parameter is s > 5, we show that in fact N (∇u, R) grows like the diameter with probability 1, albeit the ratio is not a universal constant but a random variable.

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No repulsion between critical points for planar Gaussian random fields

Cited by 7 publications

References 15 publications

Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices

Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices

On the nodal structures of random fields -- a decade of results

Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity

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