The statistics of Rayleigh-Levy flight extrema

Francis Bernardeau; Christophe Pichon

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arXiv · 2402.15915

The statistics of Rayleigh-Levy flight extrema

Francis Bernardeau·Christophe Pichon

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Abstract

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.

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Francis Bernardeau, Christophe Pichon. 2024-02-24. The statistics of Rayleigh-Levy flight extrema. https://arxiv.org/abs/2402.15915

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