Large deviations of the ballistic L\'evy walk model

TL;DR

This paper analyzes the large deviation behavior of the ballistic Lévy walk, revealing two distinct laws governing the particle density near and far from the light cone, and connecting rare events to renewal theory.

Contribution

It introduces a dual-law description of the density, combining the Lamperti-arcsine law for typical fluctuations and an infinite density for large deviations, linking large positions to the longest travel times.

Findings

The density follows the Lamperti-arcsine law away from the light cone.

Near the light cone, the density diverges, but finite-time observations remain finite.

Large deviations are characterized by an infinite density related to the longest travel time.

Abstract

We study the ballistic L\'evy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a `light' cone $-v_0 t<x<v_0 t$. In particular we study this density close to its maximum in the vicinity of the `light' cone. The spreading density follows the Lamperti-arcsine law describing typical fluctuations far from the `light' cone. However this law blows up in the vicinity of the `light' cone horizon which is nonphysical, in the sense that any finite time observation will never diverge. We claim that one can find two laws for the spatial density, the first one is the mentioned Lamperti-arcsine law describing the central part of the distribution and the second is an infinite density illustrating the dynamics for large $x$. We identify the relationship between a large position and the longest traveling time describing the single big jump principle. From the renewal theory we find that the distribution of rare events of the position is related to the derivative of the average of the number of renewals at a short `time' using a rate formalism.