We study the ballistic Lévy 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. However, this law blows up in the vicinity of the spreading 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 $x\simeq v_{0}t$. 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.

• Received 19 July 2020
• Accepted 14 October 2020

DOI:https://doi.org/10.1103/PhysRevE.102.052115