Lyapunov exponents of random walks in small random potential: the upper bound

Abstract

We consider the simple random walk on ${\mathbb{Z}}^d$ evolving in a random i.i.d. potential taking values in $[0,+\infty )$. The potential is not assumed integrable, and can be rescaled by a multiplicative factor $\lambda >0$. Completing the work started in a companion paper, we give the asymptotic behaviour of the Lyapunov exponents for $d\ge 3$, both annealed and quenched, as the scale parameter $\lambda$ tends to zero.

16 pages