The statistics of Rayleigh-Levy flight extrema

¹ Université Paris-Saclay, CNRS, CEA, Institut de physique théorique, 91191 Gif-sur-Yvette, France
² CNRS and Sorbonne Université, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, 75014 Paris, France
³ Korea Institute for Advanced Studies (KIAS), 85 Hoegi-ro, Dongdaemun-gu, Seoul 02455, Republic of Korea

Received: 16 February 2024
Accepted: 15 May 2024

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 predictions of a wide range of characteristics. Here, we derive the one- and two-point statistics for extreme points within Rayleigh-Levy flights, spanning one to three dimensions (1D–3D) and 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.