Large deviations for continuous time random walks

TL;DR

This paper derives the large deviation principles for continuous time random walks, explaining the exponential decay in particle spreading observed in complex systems, and examines how event correlations influence this behavior.

Contribution

It provides a large deviation framework for continuous time random walks under broad conditions, linking microscopic jump distributions to macroscopic spreading behaviors.

Findings

Particle spreading exhibits exponential decay with logarithmic correction.

Anti-bunching reduces exponential decay effects.

Bunching and intermittency enhance exponential decay.

Abstract

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e. L\'evy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.