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

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  • Azaïs, Jean-Marc
  • Delmas, Céline

Abstract

Let X={X(t):t∈RN} be an isotropic Gaussian random field with real values. The first part studies the mean number of critical points of X with index k using random matrices tools. An exact expression for the probability density of the kth eigenvalue of a N-GOE matrix is obtained. We deduce some exact expressions for the mean number of critical points with a given index. A second part studies the 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 proved. The correlation function between maxima (or minima) depends on the dimension of the ambient space.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:411-445
    DOI: 10.1016/j.spa.2022.04.013
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    References listed on IDEAS

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    1. Estrade, Anne & Fournier, Julie, 2016. "Number of critical points of a Gaussian random field: Condition for a finite variance," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 94-99.
    2. Azaïs, Jean-Marc & Wschebor, Mario, 2008. "A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1190-1218, July.
    3. 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.
    4. Chiani, Marco, 2014. "Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 69-81.
    5. Nicolaescu, Liviu I., 2017. "A CLT concerning critical points of random functions on a Euclidean space," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3412-3446.
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