Abstract
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.
Citation
Olivier Garet. Régine Marchand. "Large deviations for the chemical distance in supercritical Bernoulli percolation." Ann. Probab. 35 (3) 833 - 866, May 2007. https://doi.org/10.1214/009117906000000881
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