Large deviations in the reinforced random walk model on trees

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.