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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.18219 |
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| _version_ | 1866913540566155264 |
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| author | Smutek, Krzysztof |
| author_facet | Smutek, Krzysztof |
| contents | We investigate the fluctuations of the nodal number (count of the phase singularities) in a natural extension of the well-known complex planar Berry Random Wave Model - Berry (2002) - obtained by considering two independent real Berry Random Waves, with distinct energies $E_1, E_2 \to \infty$ (at possibly $\neq$ speeds). Our framework relaxes the conditions used in Nourdin, Peccati and Rossi (2019) where the energies were assumed to be identical ($E_1 \equiv E_2$). We establish the asymptotic equivalence of the nodal number with its 4-th chaotic projection and prove quantitative Central Limit Theorems (CLTs) in the 1-Wasserstein distance for the univariate and multivariate scenarios. We provide a corresponding qualitative theorem on the convergence to the White Noise in a sense of random distributions. We compute the exact formula for the asymptotic variance of the nodal number with exact constants depending on the choice of the subsequence. We provide a simple and complete characterisation of this dependency through introduction of the three asymptotic parameters: $r^{log}$, $r$, $r^{exp}$. The corresponding claims in the one-energy model were established in Nourdin, Peccati and Rossi (2019), Peccati and Vidotto (2020), Notarnicola, Peccati and Vidotto (2023), and we recover them as a special case of our results. Moreover, we establish full-correlations with polyspectra, which are analogues of the full-correlation with tri-spectrum that was previously observed for the nodal length in Vidotto (2021). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fluctuations of the Nodal Number in the Two-Energy Planar Berry Random Wave Model Smutek, Krzysztof Probability Mathematical Physics 60G60, 60B10, 60D05, 58J50, 35P20 We investigate the fluctuations of the nodal number (count of the phase singularities) in a natural extension of the well-known complex planar Berry Random Wave Model - Berry (2002) - obtained by considering two independent real Berry Random Waves, with distinct energies $E_1, E_2 \to \infty$ (at possibly $\neq$ speeds). Our framework relaxes the conditions used in Nourdin, Peccati and Rossi (2019) where the energies were assumed to be identical ($E_1 \equiv E_2$). We establish the asymptotic equivalence of the nodal number with its 4-th chaotic projection and prove quantitative Central Limit Theorems (CLTs) in the 1-Wasserstein distance for the univariate and multivariate scenarios. We provide a corresponding qualitative theorem on the convergence to the White Noise in a sense of random distributions. We compute the exact formula for the asymptotic variance of the nodal number with exact constants depending on the choice of the subsequence. We provide a simple and complete characterisation of this dependency through introduction of the three asymptotic parameters: $r^{log}$, $r$, $r^{exp}$. The corresponding claims in the one-energy model were established in Nourdin, Peccati and Rossi (2019), Peccati and Vidotto (2020), Notarnicola, Peccati and Vidotto (2023), and we recover them as a special case of our results. Moreover, we establish full-correlations with polyspectra, which are analogues of the full-correlation with tri-spectrum that was previously observed for the nodal length in Vidotto (2021). |
| title | Fluctuations of the Nodal Number in the Two-Energy Planar Berry Random Wave Model |
| topic | Probability Mathematical Physics 60G60, 60B10, 60D05, 58J50, 35P20 |
| url | https://arxiv.org/abs/2402.18219 |