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Local repulsion of planar Gaussian critical points

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  • Ladgham, Safa
  • Lachieze-Rey, Raphaël

Abstract

We study the local repulsion between critical points of a stationary isotropic smooth planar Gaussian field. We show that the critical points can experience a soft repulsion which is maximal in the case of the random planar wave model, or a soft attraction of arbitrary high order. If the type of critical points is specified (extremum, saddle point), the points experience a hard local repulsion, that we quantify with the precise magnitude of the second factorial moment of the number of points in a small ball.

Suggested Citation

  • Ladgham, Safa & Lachieze-Rey, Raphaël, 2023. "Local repulsion of planar Gaussian critical points," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:spapps:v:166:y:2023:i:c:s0304414923001850
    DOI: 10.1016/j.spa.2023.09.008
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    References listed on IDEAS

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    1. Taylor, Jonathan E. & Worsley, Keith J., 2007. "Detecting Sparse Signals in Random Fields, With an Application to Brain Mapping," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 913-928, September.
    2. Azaïs, Jean-Marc & Delmas, Céline, 2022. "Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 411-445.
    3. Muirhead, Stephen, 2020. "A second moment bound for critical points of planar Gaussian fields in shrinking height windows," Statistics & Probability Letters, Elsevier, vol. 160(C).
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