Quenched large deviations for multidimensional random walk in random environment with holding times

TL;DR

This paper establishes a quenched large deviation principle for multidimensional random walks in random environments with random holding times, providing a detailed rate function and asymptotic analysis in specific cases.

Contribution

It introduces a quenched large deviation principle for random walks with holding times, linking the rate function to Lyapunov exponents of the Laplace transform of first passage times.

Findings

Rate function given by Legendre transform of Lyapunov exponents

Derived asymptotics of the rate function in special cases

Established large deviation principles for complex random environments

Abstract

We consider a random walk in random environment with random holding times, that is, the random walk jumping to one of its nearest neighbors with some transition probability after a random holding time. Both the transition probabilities and the laws of the holding times are randomly distributed over the integer lattice. Our main result is a quenched large deviation principle for the position of the random walk. The rate function is given by the Legendre transform of the so-called Lyapunov exponents for the Laplace transform of the first passage time. By using this representation, we derive some asymptotics of the rate function in some special cases.