Subdiffusive behavior of random walk on a random cluster. (English) Zbl 0632.60106

Kesten, H.

Ann. Inst. Henri Poincaré, Probab. Stat. 22, 425-487 (1986).

Consider a particle performing a symmetric random walk on a random graph G, when G is either the family tree of a critical branching process conditioned on non-extinction, or the incipient infinite cluster of bond percolation on the two-dimensional square lattice.
In both cases the author investigates the mean square displacement of the walker, showing that it is subdiffusive in the sense that it grows at most as \(n^{\alpha}\) for \(\alpha <1\). In the former case it is shown that, when normalized by \(n^{1/3}\), the displacement of the walker has a limiting distribution.

Reviewer: G.R.Grimmet

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G50	Sums of independent random variables; random walks

Keywords:

symmetric random walk on a random graph; critical branching process; infinite cluster; percolation