Geodesics and Recurrence of Random Walks in Disordered Systems

Daniel Boivin (Université de Bretagne Sud)
Jean-Marc Derrien (Université de Bretagne Sud)

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 $\alpha > 1/2$, then the number of geodesics is either $0$ or $+\infty$. We construct a model with stationary passage times such that $E\lbrack t(e)^\alpha\rbrack < \infty$, for every $0 < \alpha < 1/2$, 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.

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Pages: 101-115

Publication Date: May 15, 2002

DOI: 10.1214/ECP.v7-1052

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