A law of large numbers concerning the number of critical points of isotropic Gaussian functions
Abstract
For any smooth random Gaussian function $\Phi$ on $\mathbb{R}^m$ we denote by $Z_N(\Phi)$ the number of critical points of $\Phi$ inside the cube $[0,N]^m$. We prove that for certain isotropic random functions $\Phi$ the ratio $N^{-m}Z_N(\Phi)$ converges a.s. and $L^2$ to a universal explicit constant $C_m(\Phi)$.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.14383
- arXiv:
- arXiv:2408.14383
- Bibcode:
- 2024arXiv240814383N
- Keywords:
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- Mathematics - Probability;
- Mathematics - Classical Analysis and ODEs;
- 60F15;
- 60D05
- E-Print:
- 26 pages, added an appendix on stationary random measures, added references