L\'{e}vy-like flights and fractal geometry of finite point sets

TL;DR

This paper investigates the fractal geometry of point sets generated by Lévy-like flights with various step distributions, developing simulation methods to analyze their correlation dimensions across scales.

Contribution

It introduces novel simulation techniques for Lévy-like flights, enabling the creation of data sets with specific fractal properties, even with few points.

Findings

Lévy-like flights can produce fractal point sets with controllable correlation dimensions.

Positive  distributions exhibit fractal behavior across a wide scale range.

Simulation methods enable analysis of fractal properties in small data sets.

Abstract

We study L\'{e}vy-like and truncated L\'{e}vy-like flights with step probability distribution of the form $r^{-1+\nu}$ for negative, positive, and zero $\nu$, focusing on the appearance of fractal geometry characteristics in the generated point sets. Forming ensembles of such point sets with fixed multiplicity, we develop simulation techniques leading to the desired value of correlation dimension in a vast continuous interval of scales. In particular, we demonstrate the possibility to produce ensembles of data sets with a low number of points with the needed properties. Furthermore, we show that the positive $\nu$ distributions, apart from a region near the upper scale limit, show fractal behaviour that extends to infinitesimally low scales. As an example, we apply our findings to producing simulations relevant to the search for critical fluctuations, related to QCD critical endpoint, in heavy-ion collision experiments.