PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

258 Geoffrey Grimmett
Theorem 11.13. Suppose that d ≥ 3. The random sets B and C are intersection-equivalent.
A similar result is valid when d = 2, but with a suitable redefinition of the
random set C. This is achieved by taking different values of p at the different
stages of the construction, namely p = k/(k + 1) at the kth stage.
This correspondence is not only beautiful and surprising, but also useful.
It provides a fairly straightforward route to certain results concerning
intersections of random walks and Brownian Motions, for example. Conversely,
using the rotation-invariance of Brownian Motion, one may obtain
results concerning projections of the random Cantor set in other directions
than onto an axis (thereby complementing results of [112], in the case of the
special parameter-value given above).
The proof of Theorem 11.13 is analytical, and proceeds by utilising
• classical potential theory for Brownian Motion,
• the relationship between capacity and percolation for trees ([250]), and
• the relationship between capacity on trees and capacity on an associated
Euclidean space ([54, 308]).
It is an attractive target to understand Theorem 11.13 via a coupling of the
two random sets.