# Limit theorems for L\'evy walks in $d$ dimensions: rare and bulk fluctuations

## TL;DR

This paper investigates the large-time behavior of d-dimensional Levy walks with heavy-tailed step durations, revealing two distinct scaling limits for the probability density function that describe bulk and tail fluctuations, including anisotropic effects.

## Contribution

It introduces a comprehensive analysis of the scaling limits of Levy walks in multiple dimensions, highlighting the anisotropic bulk distribution and the tail infinite density, extending previous one-dimensional results.

## Key findings

- The bulk of the PDF follows a generalized Levy distribution in d dimensions.
- The tail of the PDF is described by an infinite density reflecting angular structure.
- Long ballistic flights significantly influence the dispersion despite their rarity.

## Abstract

We consider super-diffusive L\'evy walks in $d \geqslant 2$ dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the $d-$dimensional generalization of the one-dimensional L\'evy distribution and is the counterpart of central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional L\'evy distribution and the CLT this distribution does not have universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of PDF of velocity in one step. Typical realization of the walk consists of anomalous diffusive motion (described by anisotropic $d-$dimensional L\'evy distribution) intermitted by long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of L\'evy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distribution the $d-$dimensional process can be reduced to one-dimensional L\'evy walk.