The statistics of Rayleigh-Levy flight extrema
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914692180475904 |
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| author | Bernardeau, Francis Pichon, Christophe |
| author_facet | Bernardeau, Francis Pichon, Christophe |
| contents | Rayleigh-Levy flights have played a significant role in cosmology as simplified models for understanding how matter distributes itself under gravitational influence. These models also exhibit numerous remarkable properties that enable the prediction of a wide range of characteristics. Here, we derive the one and two point statistics of extreme points within Rayleigh-Levy flights spanning one to three dimensions, stemming directly from fundamental principles. In the context of the mean field limit, we provide straightforward closed-form expressions for Euler counts and their correlations, particularly in relation to their clustering behaviour over long distances. Additionally, quadratures allow for the computation of extreme value number densities. A comparison between theoretical predictions in 1D and Monte Carlo measurements shows remarkable agreement. Given the widespread use of Rayleigh-Levy processes, these comprehensive findings offer significant promise not only in astrophysics but also in broader applications beyond the field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The statistics of Rayleigh-Levy flight extrema Bernardeau, Francis Pichon, Christophe Cosmology and Nongalactic Astrophysics Rayleigh-Levy flights have played a significant role in cosmology as simplified models for understanding how matter distributes itself under gravitational influence. These models also exhibit numerous remarkable properties that enable the prediction of a wide range of characteristics. Here, we derive the one and two point statistics of extreme points within Rayleigh-Levy flights spanning one to three dimensions, stemming directly from fundamental principles. In the context of the mean field limit, we provide straightforward closed-form expressions for Euler counts and their correlations, particularly in relation to their clustering behaviour over long distances. Additionally, quadratures allow for the computation of extreme value number densities. A comparison between theoretical predictions in 1D and Monte Carlo measurements shows remarkable agreement. Given the widespread use of Rayleigh-Levy processes, these comprehensive findings offer significant promise not only in astrophysics but also in broader applications beyond the field. |
| title | The statistics of Rayleigh-Levy flight extrema |
| topic | Cosmology and Nongalactic Astrophysics |
| url | https://arxiv.org/abs/2402.15915 |