Large deviations of the ballistic Lévy walk model

Area of Science:

• Physics
• Statistical Mechanics
• Non-equilibrium Systems

Background:

• Ballistic Lévy walks exhibit unique spreading dynamics.
• Infinite mean travel times lead to complex particle distributions.
• Existing models show nonphysical divergences near the spreading horizon.

Purpose of the Study:

• To investigate particle density in ballistic Lévy walks with infinite mean travel times.
• To resolve nonphysical blow-ups of the Lamperti-arcsine law near the light cone.
• To establish new laws describing particle density at the spreading horizon.

Main Methods:

• Analysis of particle density within a "light" cone (-v0t < x < v0t).
• Application of renewal theory to understand rare, large-position events.
• Investigation of the single big jump principle and its relation to travel times.

Main Results:

• Identified two distinct laws for spatial particle density.
• The Lamperti-arcsine law describes the central distribution, while a new law handles dynamics near the light cone.
• Established a connection between large positions and longest travel times.

Conclusions:

• The study resolves nonphysical divergences by introducing a dual-law description for particle density.
• The findings provide a more accurate model for particle spreading in systems with infinite mean travel times.
• Renewal theory and the single big jump principle are crucial for understanding extreme events in Lévy walks.