## Trapping in the random conductance model    [PDF]

M. Biskup, O. Louidor, A. Rozinov, A. Vandenberg-Rodes
We consider random walks on \$\Z^d\$ among nearest-neighbor random conductances
which are i.i.d., positive, bounded uniformly from above but whose support
extends all the way to zero. Our focus is on the detailed properties of the
paths of the random walk conditioned to return back to the starting point at
time \$2n\$. We show that in the situations when the heat kernel exhibits
subdiffusive decay --- which is known to occur in dimensions \$d\ge4\$ --- the
walk gets trapped for a time of order \$n\$ in a small spatial region. This shows
that the strategy used earlier to infer subdiffusive lower bounds on the heat
kernel in specific examples is in fact dominant. In addition, we settle a
conjecture concerning the worst possible subdiffusive decay in four dimensions.
View original: http://arxiv.org/abs/1202.2587