 |
|
|
| | | | | |
|
|
|
|
|
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
$Elbrack t(e)^alpharbrack
|
Full text: PDF
Pages: 101-115
Published on: May 15, 2002
|
Bibliography
-
BERGER, N. Transience, Recurrence and Critical
Behavior for Long-Range Percolation.
Preprint.
-
BREIMAN, L. (1968)
Probability,
Addison-Wesley Series in Statistics.
Addison-Wesley Publishing Company, Reading, Mass.
MR 37 #4841
-
BURTON, R. and KEANE, M. (1989)
Density and uniqueness in percolation.
Comm. Math. Phys. 121 501 - 505.
MR 90g:60090
-
DE MASI, A., FERRARI, P.A., GOLDSTEIN, S. and WICK, W.D. (1989)
An invariance principle for reversible Markov processes.
Applications to random motions in random environments.
J. Stat. Phys. 55 787-855.
MR 91e:60107
-
DOYLE, P. and SNELL, J.L. (1984)
Random walks and electrical networks.
The Carus Mathematical Monographs, 22
The Mathematical Association of America.
John Wiley & Sons, New York.
MR
-
DURRETT, R. (1986)
Multidimensional random walks in random environments
with subclassical limiting behavior,
Comm. Math. Phys. 104 87 - 102.
MR 87h:60126
-
GRIMMETT, G. (1989)
Percolation.
Springer-Verlag. New York.
MR 90j:60109
-
GRIMMETT, G.R., KESTEN, H. and ZHANG, Y. (1993)
Random walk on the infinite cluster of the percolation model.
Probab. Theory Relat. Fields 96 33-44.
MR 94i:60078
-
HAMMERSLEY, J.M. and WELSH, D.J.A. (1965) First-passage percolation,
subadditive processes, stochastic networks, and generalized renewal theory.
Bernoulli-Bayes-Laplace Anniversary Volume ,
J. Neyman and L.M. LeCam, eds. pp. 61-110,
Springer-Verlag New York, N.Y.
MR 33 #6731
-
KOZLOV, S.M. (1985)
The averaging method and walks in inhomogeneous environments.
Russ. Math. Surv. 40 73-145 ;
translation from Usp. Mat. Nauk 40 61-120.
MR 87b:60104
-
KRENGEL, U. (1985)
Ergodic Theorems.
De Gruyter Studies in Mathematics, 6. Berlin-New York: Walter de
Gruyter.
MR 87i:28001
-
LICEA, C. and NEWMAN, C. (1996)
Geodesics in two-dimensional first-passage percolation.
Ann. Probab. 24 399 - 410.
MR 97c:60238
-
LYONS, T. (1983)
A simple criterion for transience of a reversible Markov chain.
Ann. Probab. 11 393-402.
MR 84e:60102
-
NEWMAN, C. M. (1997)
Topics in disordered systems.
Lectures in Mathematics, ETH Z" urich.
Basel: Birkhäuser.
MR 99e:82052
-
PERES, Y. (1999)
Probability on trees: An introductory climb.
Bertoin, J. (ed.) et al.,
Lectures on probability theory and statistics.
Ecole d'été de Probabilités de Saint-Flour XXVII-1997,
Saint-Flour, France, July 7-23, 1997.
Berlin: Springer. Lect. Notes Math. 1717, 193-280.
MR 2001c:60139
-
WEHR, J. (1997) On the Number of Infinite
Geodesics and Ground States in Disordered Systems.
J. Stat. Physics 87 439 -447.
MR 98k:82087
|
|
|
|
|
|
|
|
| | | | |
Electronic Communications in Probability. ISSN: 1083-589X |
|
Context