Geodesics and Recurrence of Random Walks in Disordered Systems

Abstract

In a first-passage percolation model on the square lattice $Z^2$, if the passage times are independent then the number of geodesics is either $0$ or $+\infty$. If the passage times are stationary, ergodic and have a finite moment of order $\alpha \gt 1/2$, then the number of geodesics is either $0$ or $+\infty$. We construct a model with stationary passage times such that $E\lbrack t(e)^\alpha\rbrack \lt \infty$, for every $0 \lt \alpha \lt 1/2$, and with a unique geodesic. The recurrence/transience properties of reversible random walks in a random environment with stationary conductances $( a(e);e$ is an edge of $\mathbb{Z}^2)$ are considered.