Functional CLT for random walk among bounded random conductances. (English) Zbl 1127.60093

Biskup, Marek; Prescott, Timothy M.

Electron. J. Probab. 12, 1323-1348 (2007).

Summary: We consider the nearest-neighbor simple random walk on \(\mathbb Z^{d}\), \(d\geq 2\), driven by a field of i.i.d. random nearest-neighbor conductances \(\omega _{xy}\in [0,1]\). Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the \(\omega \)’s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in \(d\geq 5\) due to anomalously slow decay of the probability that the walk returns to the starting point at a given time.

MSC:

60K37	Processes in random environments
60F05	Central limit and other weak theorems
82C41	Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

Keywords:

heat kernel; percolation; isoperimetry; random nearest-neighbor conductances