PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

224 Geoffrey Grimmett
9. PERCOLATION IN TWO DIMENSIONS
9.1 The Critical Probability is 1
2
The famous exact calculation for bond percolation on L 2 is the following,
proved originally by Kesten [201]. The proof given here is taken from [G].
Theorem 9.1. The critical probability of bond percolation on Z2 equals 1
2 .
Furthermore, θ( 1
2 ) = 0.
Proof. Zhang discovered a beautiful proof that θ( 1
2 ) = 0, using only the
uniqueness of the infinite cluster. Set p = 1
2 . Let T(n) be the box T(n) =
[0, n] 2 , and find N sufficiently large that
� � 1
P1 ∂T(n) ↔ ∞ > 1 −
2
84 for n ≥ N.
We set n = N + 1. Writing A l , A r , A t , A b for the (respective) events that the
left, right, top, bottom sides of T(n) are joined to ∞ off T(n), we have by
the FKG inequality that
� �
P1 T(n) � ∞ = P1
2
2 (Al ∩ Ar ∩ At ∩ Ab )
by symmetry, for g = l,r,t,b. Therefore
≥ P1
2 (Al )P(A r )P(A t )P(A b )
= P1
2 (Ag ) 4
P1
2 (Ag � � �
) ≥ 1 − 1 − P1 T(n) ↔ ∞
2
�1/4 > 7
8 .
) : 0 ≤
x1, x2 < n}. Let Al d , Ar d , At d , Ab d denote the (respective) events that the left,
right, top, bottom sides of T(n)d are joined to ∞ by a closed dual path off
T(n)d. Since each edge of the dual is closed with probability 1
2 , we have that
Now we move to the dual box, with vertex set T(n)d = {x + ( 1
2
2 (Ag 7
d ) > 8
P1
for g = l,r,t,b.
Consider the event A = Al ∩ Ar ∩ At d ∩ Ab d , and see Figure 9.1. Clearly
1
1
P1 (A) ≤
2 2 , so that P1 (A) ≥
2 2 . However, on A, either L2 has two infinite
open clusters, or its dual has two infinite closed clusters. Each event has
probability 0, a contradiction. We deduce that θ( 1
2 ) = 0, implying that
pc ≥ 1
2 .
Next we prove that pc ≤ 1
2 . Suppose instead that pc > 1
2 , so that
� � −γn
(9.2) P1 0 ↔ ∂B(n) ≤ e for all n,
2
, 1
2