PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT
PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT
PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT
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Hence,<br />
Percolation and Disordered Systems 219<br />
�<br />
Pp 0 ↔ x, u ↔ y off CB(x) � ≥ Ep<br />
We have proved that<br />
�<br />
Pp 0 ↔ x, u ↔ y off CB(x) � ≥<br />
= Ep<br />
�<br />
�<br />
Ep 1{0↔x}τ CB(x)<br />
p<br />
�<br />
1{0↔x}τ CB(x)<br />
p<br />
� �<br />
τp(0, x)τp(u, y) − Ep 1{0↔x}τp(u, y) � − Ep<br />
Applying (8.20a), we have that<br />
τp(u, y) − τ CB(x)<br />
p (u, y) ≤ �<br />
w∈CB(x)<br />
(u, y) � .<br />
(u, y) � �CB(x) ��<br />
�<br />
1{0↔x}τ CB(x)<br />
p (u, y) ��<br />
.<br />
τp(u, w)τp(y, w),<br />
whence<br />
�<br />
(8.20c) Pp 0 ↔ x, u ↔ y off CB(x) � ≥<br />
τp(0, x)τp(u, y) − � � �<br />
Pp 0 ↔ x, w ↔ x off B τp(u, w)τp(y, w).<br />
w∈Z d \B<br />
Finally, using the BK inequality,<br />
� � �<br />
Pp 0 ↔ x, w ↔ x off B ≤<br />
v∈Z d \B<br />
τp(0, v)τp(w, v)τp(x, v).<br />
We insert this into (8.20c), and deduce via (8.20b) that<br />
(8.20)<br />
(8.21)<br />
By (8.12),<br />
dχ<br />
dp<br />
≥ 2dα(p)χ2<br />
�<br />
1 − sup<br />
�<br />
|u|=1<br />
v,w/∈B<br />
�<br />
τp(0, v)τp(v, w)τp(w, u) .<br />
�<br />
τp(0, v)τp(v, w)τp(w, u) ≤ T(p) for all u.<br />
v,w<br />
Assuming that T(pc) < ∞, we may choose B = B(R) sufficiently large that<br />
(8.22)<br />
dχ<br />
dp ≥ 2dα(p)χ2 (1 − 1<br />
2 ) for p ≤ pc.<br />
Integrate this, as for (8.15), to obtain that<br />
χ(p) ≤<br />
1<br />
α ′ (pc − p)<br />
for p ≤ pc