Effective parameters of random walks on critical random graphs
Final Report Abstract
This project investigates random walks on random graphs. Of particular interest was the long-term behavior and parameters such as the effective speed and the diffusivity when the underlying graph is close to a critical state, in which it exhibits a complicated, fractal-like structure. It is well-known that simple random walk on a supercritical percolation cluster converges under diffusive rescaling to a Brownian motion. On a critical version of the percolation cluster, the incipient infinite cluster, the random walk is subdiffusive, so one would expect the diffusivity to vanish as the cluster becomes critical. Together with Remco van der Hofstad and Tim Hulshof, we investigated this transition in a mean-field model, for a random walk on a branching random walk. We could show that as time tends to infinity and simultaneously the underlying braching random walk approaches criticality, the random walk on the branching random walk converges to a Brownian motion. The main feature of this convergence is that we can in some sense interpolate between the supercritical and the critical regime, depending on how fast the branching process becomes critical, for example in terms of the scaling factors. We could also show that the effective diffusivity of a random walk on a branching random walk vanishes as the branching process becomes critical, with the same order as is conjectured in the percolation case. With Tim Hulshof, we also considered the diffusivity in the percolation case, and studied its behavior as the percolation cluster becomes critical. We could show that the diffusivity indeed vanishes and give an upper polynomial bound.
Publications
- Random walk on barely supercritical branching random walk
R. van der Hofstad, T. Hulshof, J. Nagel
(See online at https://doi.org/10.1007/s00440-019-00942-0)
