Invasion percolation on regular trees

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behaviour of its connectivity functions, and of its volume both at a given height and below a given height. We find that the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, but the scaling functions differ. Thus, somewhat surprisingly, the invasion percolation cluster and the incipient infinite cluster are globally different. However, far above the root, the two clusters do have the same law locally.

In addition, we use recent work of Barlow, Jarai, Kumagai and Slade to analyse simple random walk on the invasion percolation cluster, and show that the spectral dimension is 4/3, as it is on the incipient infinite cluster.

This is joint work with Omer Angel, Jesse Goodman and Frank den Hollander.