Proof. Take K such that 2
K
≤ Lp 2
K+1
and ˆ P
t
the linear interpolation between P
1 2
and P
p
. By gluing arguments, for A =
{0 ∂ S
Lp
}, for any v ∈ S
2
K −4
,2
K −3
, ˜
P
t
v is pivotal for A ≥ C
1
˜ P
t
∂ S
2
K −5
˜ P
t
∂ S
2
K −2
∂ S
Lp
˜ P
t
v
4, σ
4
∂ S
2
K −5
v ≥ C
2
˜ P
t
∂ S
2
K
˜ P
t
v
4, σ
4
∂ S
2
K −5
v ,
so that d
d t log
˜ P
t
A ≥
X
v ∈S
2K−4,2K−3
p − 12ˆP
t
v
4, σ
4
∂ S
2
K −5
v ≥ C
3
p − 12Lp
2
ˆ P
t 4,
σ
4
∂ S
Lp
, since each of the sites v
∈ S
2
K −4
,2
K −3
produces a contribution of order ˆ P
t 4,
σ
4
∂ S
Lp
. Proposi- tion 34, proved later
4
, allows to conclude.
7 Consequences for the characteristic functions
7.1 Different characteristic lengths
Roughly speaking, a characteristic length is a quantity intended to measure a “typical” scale of the system. There may be several natural definitions of such a length, but we usually expect the different
possible definitions to produce lengths that are of the same order of magnitude. For two-dimensional percolation, the three most common definitions are the following:
Finite-size scaling
The lengths L
ε
that we have used throughout the paper, introduced in [15], are known as “finite-size scaling characteristic lengths”:
L
ε
p = min
{n s.t. P
p
C
H
[0, n] × [0, n] ≤ ε} when p 12, min
{n s.t. P
p
C
∗ H
[0, n] × [0, n] ≤ ε} when p 12. 7.1
Mean radius of a finite cluster
The quadratic mean radius measures the “typical” size of a finite cluster. It can be defined by the formula
ξp =
1 E
p
|C0|; |C0| ∞ X
x
kxk
2 ∞
P
p
0 x, |C0| ∞
1 2
. 7.2
4
This does not raise any problem since we have included this complementary bound only for the sake of completeness, and we will not use it later.
1604
Connection probabilities
A third possible definition would be via the rate of decay of correlations. Take first p 12 for
example. For two sites x and y, we consider the connection probability between them τ
x, y
:= P
p
x y ,
7.3 and then
τ
n
:= sup
x ∈∂ S
n
τ
0,x
, 7.4
the maximum connection probability between sites at distance n using translation invariance. For any n, m
≥ 0, we have τ
n+m
≥ τ
n
τ
m
, in other words
− log τ
n n
≥0
is sub-additive, which implies the existence of a constant ˜ ξp such
that −
log τ
n
n −→
1 ˜
ξp = inf
m
− log
τ
m
m 7.5
when n → ∞. Note the following a-priori bound:
P
p
0 x ≤ e
−kxk
∞
˜ ξp
. 7.6
For p 12, we simply use the symmetry p ↔ 1 − p: we consider
τ
∗ n
:= sup
x ∈∂ S
n
P
p ∗
x 7.7
and then ˜ ξp in the same way. We have in this case
P
p ∗
x ≤ e
−kxk
∞
˜ ξp
. 7.8
Note that the symmetry p ↔ 1 − p gives immediately
˜ ξp = ˜
ξ1 − p.
Relation between the different lengths
As expected, these characteristic lengths turn out to be all of the same order of magnitude: we will prove in Section 7.3 that L
ε
≍ L
ε
′
for any two ε, ε
′
∈ 0, 12, in Section 7.4 that L ≍ ˜ ξ, and in
Section 7.5 that L ≍ ξ.
7.2 Main critical exponents
