Project summary

 

We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Additionally, we analyze the forward maximal weights along the backbone of the invasion cluster and prove a limit law for the process.

 

In doing so we encounter the quenched survival function, g(T,p) := probability that the random tree T survives p-Bernoulli percolation, conditioned on T.  We show that this is almost surely smooth on (p_c,1) and, provided the offspring distribution has enough moments, the derivatives extend continuously to p_c.

 

Papers

Michelen, M; Pemantle, R; Rosenberg, J. (2018) Invasion Percolation on Galton Watson Trees

Michelen, M; Pemantle, R; Rosenberg, J. (2018) Quenched survival of Bernoulli percolation on Galton-Watson trees.

 Problems

  1. Can the assumption on P[Z = 1] for absolute continuity of the invasion measure be entirely removed?
  2. Can we prove that the two measures are equivalent?  In other words, do we have that the Limit Uniform measure is absolutely continuous w.r.t. the Invasion Measure?
  3. Can the moment conditions for higher order differentiability of the quenched survival function at criticality be made sharp?