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**Invasion percolation on regular trees.**
*(English)*
Zbl 1145.60050

The authors study the scaling limit of the Infinite Percolation Cluster (IPC) and compare it to this of the Infinite Incipient Cluster (IIC), both on rooted trees, and provide in particular a negative answer to a recent conjecture claiming that their scaling limits should be the same. They indeed show here that, on trees, IPC does not have the already known scaling limit of the ICC and even more, that these two different limit laws were mutually singular. while they are also proved to look identical locally. IPC is also proved to be stochastically dominated by the IIC. Addictionally, the author prove scaling estimates for the Simple Random Walk on the IPC starting from the root, establishing that the IPC has a spectral 4/3 equal to this of the IIC. According to the authors, there is no reason to believe that these different scaling limits will disappear for other graphs, such as \(Z^d\), on which the already mentioned conjecture is expected to be false.

Reviewer: Arnaud Le Ny (Orsay)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B43 | Percolation |

### Keywords:

invasion percolation cluster; incipient infinite cluster; r-point function; cluster size; simple random walk; Poisson point process
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\textit{O. Angel} et al., Ann. Probab. 36, No. 2, 420--466 (2008; Zbl 1145.60050)

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