Fluctuations of the total number of critical points of random spherical harmonics

Cammarota, Valentina; Wigman, Igor

doi:10.1016/j.spa.2017.02.013

Cited by 27 publications

(51 citation statements)

References 17 publications

(33 reference statements)

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“…s Note that also in this case, the second component is the leading term of the expansion and it is important to stress how the leading terms in the variances cancel in all cases at the threshold u = −∞; in other words, the variance is smaller when we focus on the total number of critical points (see Cammarota & Wigman, 2017). This is again a form of the so-called "Berry's cancellation phenomenon", which we have also discussed earlier for the Lipschitz-Killing curvatures.…”

Section: F I G U R Ementioning

confidence: 99%

“…Because this second‐order chaos component (and hence the leading term in the variance) vanishes at

u = - \infty, 0

, the next component becomes of interest; it can be shown that this term is proportional to the fourth‐order chaos, and indeed, for the total number of critical points, it holds that (see Cammarota & Wigman, ).

\begin{matrix} E false[N_{- \infty}^{c} (f_{ℓ}) false] = \frac{2 ℓ (ℓ + 1)}{\sqrt{3}} + O false(1 false), \\ Var false(N_{- \infty}^{c} (f_{ℓ}) false) = \frac{ℓ^{2} \log ℓ}{27 π^{2}} + O false(ℓ^{2} false); \end{matrix}

moreover, it is also possible to consider separately extrema (minima and maxima) and saddles, yielding

\begin{matrix} E false[N_{- \infty}^{e} (f_{ℓ}) false] = \frac{ℓ (ℓ + 1)}{\sqrt{3}} + O false(1 false), \\ Var false(N_{- \infty}^{e} (f_{ℓ}) false) = \frac{ℓ^{2} \log ℓ}{4 \times 27 π^{2}} + O false(ℓ^{2} false), \end{matrix}

and

\begin{matrix} E false[N_{- \infty}^{s} (f_{ℓ}) false] = \frac{ℓ (ℓ + 1)}{\sqrt{3}} + O false(1 false), \\ Var false(N<...> \end{matrix}

…”

Section: Characterization Of Critical Points For Random Spherical Harmentioning

confidence: 99%

Section: Characterization Of Critical Points For Random Spherical Harmentioning

confidence: 99%

“…As a further tool of investigation, we shall consider in this paper also the behavior of critical points for random spherical harmonics, which has recently been fully characterized by Cammarota and Marinucci (2018b), Cammarota et al (2016b), Cammarota and Wigman (2017), among others. More precisely, by definition critical points, extrema and saddles are, respectively, given by: where we used a = c, e, s to label critical points, extrema, and saddles respectively.…”

Section: Critical Points For Random Spherical Harmonicsmentioning

confidence: 99%

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A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

et al. 2019

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A lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high‐frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz‐Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler‐Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so‐called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data.

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Section: F I G U R Ementioning

confidence: 99%

“…Because this second‐order chaos component (and hence the leading term in the variance) vanishes at

u = - \infty, 0

\begin{matrix} E false[N_{- \infty}^{c} (f_{ℓ}) false] = \frac{2 ℓ (ℓ + 1)}{\sqrt{3}} + O false(1 false), \\ Var false(N_{- \infty}^{c} (f_{ℓ}) false) = \frac{ℓ^{2} \log ℓ}{27 π^{2}} + O false(ℓ^{2} false); \end{matrix}

moreover, it is also possible to consider separately extrema (minima and maxima) and saddles, yielding

\begin{matrix} E false[N_{- \infty}^{e} (f_{ℓ}) false] = \frac{ℓ (ℓ + 1)}{\sqrt{3}} + O false(1 false), \\ Var false(N_{- \infty}^{e} (f_{ℓ}) false) = \frac{ℓ^{2} \log ℓ}{4 \times 27 π^{2}} + O false(ℓ^{2} false), \end{matrix}

and

\begin{matrix} E false[N_{- \infty}^{s} (f_{ℓ}) false] = \frac{ℓ (ℓ + 1)}{\sqrt{3}} + O false(1 false), \\ Var false(N<...> \end{matrix}

…”

Section: Characterization Of Critical Points For Random Spherical Harmentioning

confidence: 99%

Section: Characterization Of Critical Points For Random Spherical Harmentioning

confidence: 99%

Section: Critical Points For Random Spherical Harmonicsmentioning

confidence: 99%

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A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

et al. 2019

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“…This relates to the broad effort of understanding the statistical structure of stationary points (minima, maxima and saddles) of random landscapes which is of steady interest in theoretical physics [30][31][32][33][34][35][36][37][38][39], with recent applications to statistical physics [10,[34][35][36][38][39][40][41], neural networks and complex dynamics [42][43][44][45][46], string theory [47,48] and cosmology [49,50]. It is also of active current interest in pure and applied mathematics [51][52][53][54][55][56][57][58][59][60], For the model (1)- (2) in the simplest case d = 0 (x is a single point), the mean number of stationary points and of minima of the energy function was investigated in the limit of large N 1 in [35,38,39], see also [37,50,52]. It was found that a sharp transition occurs from a 'simple' landscape for µ > µ c (the same µ...…”

Section: Motivation and Goals Of The Papermentioning

confidence: 99%

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

Fyodorov

Doussal

2020

J Stat Phys

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We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature µ. We are interested in the mean spectral density ρ(λ) of the Hessian matrix K at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for ρ(λ) for a fixed L d in the N → ∞ limit extending d = 0 results of our previous work [11]. A particular attention is devoted to analyzing the limit of extended lattice systems by letting L → ∞. In all cases we show that for a confinement curvature µ exceeding a critical value µ c , the so-called "Larkin mass", the system is replicasymmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at µ → µ c . For µ < µ c the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all d as (µ c − µ) 4 near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called "marginal cases" in d = 1, 2 which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning. arXiv:1903.07159v1 [cond-mat.dis-nn]

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Monochromatic Random Waves for General Riemannian Manifolds

Canzani

2020

Frontiers in Analysis and Probability

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Fluctuations of the total number of critical points of random spherical harmonics

Cited by 27 publications

References 17 publications

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

Monochromatic Random Waves for General Riemannian Manifolds

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