Geodesics and Recurrence of Random Walks in Disordered Systems

@article{Boivin2002GeodesicsAR,
  title={Geodesics and Recurrence of Random Walks in Disordered Systems},
  author={Daniel J. Boivin and Jean-Marc Derrien},
  journal={Electronic Communications in Probability},
  year={2002},
  volume={7},
  pages={101-115}
}
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… Expand
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