## Invasion percolation on regular trees

- Date: 11/01/2006

Gordon Slade (UBC)

University of British Columbia

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.

Probability Seminar 2006