Tim Hulshof

Random walk on barely supercritical branching random walk

Random walk on a d-dimensional supercritically percolated lattice has Brownian motion as its scaling limit. When the percolation is critical, random walk behaves subdiffusively, and the scaling limit (if it exists) must have dramatically different properties. It is, however, not clear whether this subdiffusive behavior is a natural phenomenon, or due to the pathological construction of the process.

To investigate this, we consider an easier but related model: We consider supercritical Galton-Watson tree conditioned to survive, embedded into the d-dimensional lattice according to a simple random walk step distribution. We percolate the tree with parameter p. We consider a random walk on the percolated tree, and investigate the behavior of the process embedded into the lattice. We take the simultaneous limit of letting the number of steps tend to infinity, and the percolation parameter tend to the critical point. We show that the resulting scaling limit for the embedded process interpolates between the diffusive and subdiffusive regimes.

Joint work with Remco van der Hofstad and Jan Nagel.