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.

Other Information: 

Probability Seminar 2006