Large deviations in the reinforced random walk model on trees

TL;DR

This paper investigates large deviation behaviors of linearly and once-reinforced random walks on trees, revealing different tail decay rates and providing a detailed probabilistic analysis of their upper and lower tail deviations.

Contribution

It establishes the large deviation principles for both models and characterizes the decay rates of their tail probabilities in the transient phase.

Findings

Large deviations for upper tails in both models

Exponential decay for lower tail in once-reinforced model

Polynomial decay for lower tail in linearly reinforced model

Abstract

In this paper, we consider the linearly reinforced and the once-reinforced random walk models in the transient phase on trees. We show the large deviations for the upper tails for both models. We also show the exponential decay for the lower tail in the once-reinforced random walk model. On the other hand, the lower tail is in polynomial decay for the linearly reinforced random walk model.