# 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}
}
• Published 2002
• Mathematics
• Electronic Communications in Probability
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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#### References

SHOWING 1-10 OF 18 REFERENCES
Transience, Recurrence and Critical Behavior¶for Long-Range Percolation
Abstract: We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d= 1,2, where x and y are connected with probability . We show that ifExpand
On the number of infinite geodesics and ground states in disordered systems
We study first-passage percolation models and their higher dimensional analogs—models of surfaces with random weights. We prove that under very general conditions the number of lines or, in theExpand
Density and uniqueness in percolation
• Mathematics
• 1989
Two results on site percolation on thed-dimensional lattice,d≧1 arbitrary, are presented. In the first theorem, we show that for stationary underlying probability measures, each infinite cluster hasExpand
An invariance principle for reversible Markov processes. Applications to random motions in random environments
• Physics
• 1989
We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in randomExpand
Random walk on the infinite cluster of the percolation model
• Mathematics
• 1993
SummaryWe consider random walk on the infinite cluster of bond percolation on ℤd. We show that, in the supercritical regime whend≧3, this random walk is a.s. transient. This conclusion is achieved byExpand
Multidimensional random walks in random environments with subclassical limiting behavior
In this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have theExpand
The method of averaging and walks in inhomogeneous environments
CONTENTSIntroductionChapter I. General properties of walks on an inhomogeneous lattice ??1. The invariant measure and the law of large numbers ??2. The Lindeberg-Brown theorem and its corollariesExpand
Random Walks and Electrical Networks
• Computer Science
• 1984
An intriguing connection between random walks and electricity flow in networks is presented and the notion of effective resistance between two nodes in a graph is introduced, which has found several applications recently, including in graph sparsification. Expand
Geodesics in two-dimensional first-passage percolation
• Mathematics
• 1996
We consider standard first-passage percolation on Z 2 . Geodesics are nearest-neighbor paths in Z 2 , each of whose segments is time-minimizing. We prove part of the conjecture that doubly infiniteExpand
First-Passage Percolation, Subadditive Processes, Stochastic Networks, and Generalized Renewal Theory
• Mathematics
• 1965
In 1957, Broadbent and Hammersley gave a mathematical formulation of percolation theory. Since then much work has been done in this field and has now led to first-passage percolation problems. In theExpand