Large deviations for the chemical distance in supercritical Bernoulli percolation
Garet, Olivier ; Marchand, Régine
Ann. Probab., Tome 35 (2007) no. 1, p. 833-866 / Harvested from Project Euclid
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The chemical distance D(x, y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behavior of this random metric, and we prove that, for an appropriate norm μ depending on the dimension and the percolation parameter, the probability of the event ¶ \[\biggl\{\ 0\leftrightarrow x,\frac{D(0,x)}{\mu(x)}\notin (1-\varepsilon ,1+\varepsilon )\ \biggr\}\] ¶ exponentially decreases when ‖x‖1 tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.
Publié le : 2007-05-14
Classification:  Percolation,  first-passage percolation,  chemical distance,  shape theorem,  large deviation inequalities,  60K35,  82B43 
@article{1178804316,
     author = {Garet, Olivier and Marchand, R\'egine},
     title = {Large deviations for the chemical distance in supercritical Bernoulli percolation},
     journal = {Ann. Probab.},
     volume = {35},
     number = {1},
     year = {2007},
     pages = { 833-866},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1178804316}
}

Garet, Olivier; Marchand, Régine. Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Probab., Tome 35 (2007) no. 1, pp.  833-866. http://gdmltest.u-ga.fr/item/1178804316/