Mathematics – Probability
Scientific paper
2006-08-04
Annals of Probability 2008, Vol. 36, No. 2, 420-466
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/07-AOP346 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
10.1214/07-AOP346
We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its $r$-point function for any $r\geq2$ and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.
Angel Omer
Goodman Jesse
Hollander Frank den
Slade Gordon
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