Chapter 7 Infinite product spaces

7.5. TWO ZERO-ONE LAWS 15
Corollary 7.23 For each d 1 there exists pc 2 [0, 1] such that
(
0, if p < pc,
Pp(E•) =
1, if p > pc,
At p = pc we have Ppc(E•) 2{0, 1}.
(7.43)
Proof. We have already shown that E• 2 T and so Pp(E•) 2{0, 1} for all p. It
is also clear that the larger p the more edges there are and so Pp(E•) should be
increasing. However, to prove this we have to work a bit.
The key idea is that we can actually realize percolation for all p’s on the same probability
space. Consider i.i.d. random variables U =(U b) b2B whose law is uniform
on [0, 1] and define
h (p)
b = 1 {U bapplep}. (7.44)
Clearly, h (p) =(h (p)
b ) are i.i.d. Bernoulli(p) so they realize a sample of percolation at
parameter p. Now let E•(p) ={U : h (p) contains infinite connected component}.
Since p 7! h (p)
b increases, if E•(p) occurs, then so does E•(p 0 ) for p0 > p. In other
words
p 0 > p ) E•(p) ⇢ E•(p 0 ) (7.45)
This implies
Pp(E•) =P E•(p) apple P E•(p 0 ) = P p 0(E•), (7.46)
i.e., p 7! Pp(E•) is non-decreasing. Since it takes values zero or one, there exists a
unique point where Pp(E•) jumps from zero to one. This defines pc.
Let us remark what really happens: It is known that
(
= 1, d = 1,
pc
2 (0, 1), d 2.
(7.47)
At d = 2 it is known that pc = 2 (Kesten’s Theorem) but the values for d 3
are not known explicitly (and presumably are not of any special form). Concerning
Ppc(E•), it is widely believed that this probability is zero—there is no infinite
cluster at the critical point—but the proof exists only in d = 2 and d 19.
Our next object of interest is the random variable:
N = number of infinite connected components in h (7.48)
The values N can take are {0, 1, . . . }[{•}. The random variable N definitely
depends on any finite number of edges and so N is not T -measurable. (The reason
why we care is if it were tail measurable then it would have to be constant a.s.)
However, N is clearly translation invariant in the following sense:
Definition 7.24 Let tx : {0, 1} B !{0, 1} B be the map defined by
(txh) b = h b+x
(7.49)
We call tx the translation by x. An event A is translation invariant if t 1
x (A) =A for
all x 2 Z d . A random variable N is translation invariant if N tx = N.