PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

232 Geoffrey Grimmett
10. RANDOM WALKS IN RANDOM LABYRINTHS
10.1 Random Walk on the Infinite Percolation Cluster
It is a classical result that symmetric random walk on L d is recurrent when
d = 2 but transient when d ≥ 3 (see [169], pages 188, 266). Three-dimensional
space is sufficiently large that a random walker may become lost, whereas
two-dimensional space is not. The transience or recurrence of a random walk
on a graph G is a crude measure of the ‘degree of connectivity’ of G, a
more sophisticated measure being the transition probabilities themselves. In
studying the geometry of the infinite open percolation cluster, we may ask
whether or not a random walk on this cluster is recurrent.
Theorem 10.1. Suppose p > pc. Random walk on the (a.s. unique) infinite
open cluster is recurrent when d = 2 and a.s. transient when d ≥ 3.
This theorem, proved in [163] 6 , follows by a consideration of the infinite
open cluster viewed as an electrical network. The relationship between random
walks and electrical networks is rather striking, and has proved useful
in a number of contexts; see [117].
We denote the (a.s.) unique infinite open cluster by I = I(ω), whenever it
exists. On the graph I, we construct a random walk as follows. First, we set
S0 = x where x is a given vertex of I. Given S0, S1, . . . , Sn, we specify that
Sn+1 is chosen uniformly from the set of neighbours of Sn in I, this choice
being independent of all earlier choices. We call ω a transient configuration if
the random walk is transient, and a recurrent configuration otherwise. Since
I is connected, the transience or recurrence of S does not depend on the
choice of the starting point x.
The corresponding electrical network arises as follows. For x ∈ I, we
denote by Bn(x) the set of all vertices y of I such that δ(x, y) ≤ n, and we
write ∂Bn(x) = Bn(x)\Bn−1(x). We turn Bn(x) into a graph by adding
all induced (open) edges of I. Next we turn this graph into an electrical
network by replacing each edge by a unit resistor, and by ‘shorting together’
all vertices in ∂Bn(x). Let Rn(x) be the effective resistance of the network
between x and the composite vertex ∂Bn(x).
By an argument using monotonicity of effective resistance (as a function
of the individual resistances), the increasing limit R∞(x) = limn→∞ Rn(x)
exists for all x. It is a consequence of the relationship between random walk
and electrical networks that the random walk on I, beginning at x, is transient
if and only if R∞(x) < ∞. Therefore Theorem 10.1 is a consequence of the
following.
6 For a quite different and more recent approach, see [53].