PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

222 Geoffrey Grimmett
The infra-red bound emerges in the proof of (c), of which there follows an
extremely brief account.
We write x ⇔ y, and say that x is ‘doubly connected’ to y, if there exist
two edge-disjoint open paths from x to y. We express τp(0, x) in terms of
the ‘doubly connected’ probabilities δp(u, v) = Pp(u ⇔ v). In doing so, we
encounter formulae involving convolutions, which may be treated by taking
transforms. At the first stage, we have that
�
�
{0 ↔ x} = {0 ⇔ x} ∪
〈u,v〉
� 0 ⇔ (u, v) ↔ x � �
where � 0 ⇔ (u, v) ↔ x � represents the event that 〈u, v〉 is the ‘first pivotal
edge’ for the event {0 ↔ x}, and that 0 is doubly connected to u. (Similar
but more complicated events appear throughout the proof.) Therefore
(8.27) τp(0, x) = δp(0, x) + �
〈u,v〉
Now, with A(0, u; v, x) = {v ↔ x off C 〈u,v〉(0)},
� �
Pp 0 ⇔ (u, v) ↔ x .
� �
Pp 0 ⇔ (u, v) ↔ x
� �
= pPp 0 ⇔ u, A(0, u; v, x)
� � �
= pδp(0, u)τp(v, x) − pEp 1 {0⇔u} τp(v, x) − 1A(0,u;v,x) �
whence, by (8.27),
(8.28) τp(0, x) = δp(0, x) + δp ⋆ (pI) ⋆ τp(x) − Rp,0(0, x)
where ⋆ denotes convolution, I is the nearest-neighbour function I(u, v) = 1
if and only if u ∼ v, and Rp,0 is a remainder.
Equation (8.28) is the first step of the lace expansion, In the second step,
the remainder Rp,0 is expanded similarly, and so on. Such further expansions
yield the lace expansion: if p < pc then
(8.29) τp(0, x) = hp,N(0, x) + hp,N ⋆ (pI) ⋆ τp(x) + (−1) N+1 Rp,N(0, x)
for appropriate remainders Rp,N, and where
hp,N(0, x) = δp(0, x) +
N�
(−1) j Πp,j(0, x)
and the Πp,n are appropriate functions (see Theorem 4.2 of [180]) involving
nested expectations of quantities related to ‘double connections’.
j=1