The topology of Rayleigh-Levy flights in two dimensions

¹ Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Hannover 30167, Germany
² Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
³ Department of Physics, POSTECH, Pohang 37673, Republic of Korea
⁴ Université Paris-Saclay, CNRS, CEA, Institut de physique théorique, 91191 Gif-sur-Yvette, France
⁵ Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, F-75014 Paris, France
⁶ Kyung Hee University, Dept. of Astronomy & Space Science, Yongin-shi, Gyeonggi-do 17104, Republic of Korea

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 4 June 2025
Accepted: 27 October 2025

Abstract

Rayleigh-Lévy flights are simplified cosmological tools that capture certain essential statistical properties of the cosmic density field, including hierarchical structures in higher-order correlations, making them a valuable reference for studying the highly non-linear regime of structure formation. Unlike standard Markovian processes, they exhibit long-range correlations at all orders. Following on from recent work on 1D flights, this study explores the one-point statistics and Minkowski functionals (density probability distribution function, perimeter, Euler characteristic) of Rayleigh-Lévy flights in two dimensions. We derive the Euler characteristic in the mean field approximation and the density PDF and iso-field perimeter, W₁, in beyond-mean-field calculations, and validate the results against simulations. The match is excellent throughout, even for fields with large variances, in particular when finite volume effects in the simulations are taken into account and when the calculation is extended beyond the mean field.