5Scaling limit of critical percolation}\label{sec:percolation
III Schramm--Loewner Evolutions
5 Scaling limit of critical percolation
The reason we care about SLE is that they come from scaling limits of certain
discrete models. In this chapter, we will “show” that
SLE
6
corresponds to the
scaling limit of critical percolation on the hexagonal lattice.
Let
D ⊆ C
be simply connected, and
x, y ∈ ∂D
distinct. Pick a hexagonal
lattice in
D
with hexagons of size
ε
such that
x
and
y
are lattice points. We
perform critical percolation on the hexagonal lattice, i.e. for each hexagon, we
colour it black or white with probability
p
=
1
2
. We enforce the condition that
the hexagons that intersect the clockwise arc of
∂D
from
x
to
y
to all be black,
and those along the counterclockwise arc must be white.
x
y
Then there exists a unique interface
γ
ε
that connects
x
to
y
with the property
that the black hexagons on its left and white on its right. It was conjectured
(and now proved by Smirnov) that the limit of the law of
γ
ε
exists in distribution
and is conformally invariant.
This means that if
˜
D
is another simply connected domain, and
˜x, ˜y ∈ ∂
˜
D
are
distinct, then
ϕ
:
D →
˜
D
is a conformal transformation with
ϕ
(
x
) =
˜x
,
ϕ
(
y
) =
˜y
,
then
ϕ
(
γ
) is equal in distribution of the scaling limit of percolation in
˜
D
from
˜x
to ˜y.
Also, percolation also satisfies a natural Markov property: if we condition on
γ
ε
up to a given time
t
, then the rest of
γ
ε
is a percolation exploration in the
remaining domain. The reason for this is very simple — the path only determines
the colours of the hexagons right next to it, and the others are still randomly
distributed independently.
If we assume these two properties, then the limiting path
γ
satisfies the
conformal Markov property, and so must be an
SLE
κ
. So the question is, what
is κ?
To figure out this
κ
, we observe that the scaling limit of percolation has a
locality property, and we will later see that
SLE
6
is the only SLE that satisfies
this locality property.
To explain the locality property, fix a simply-connected domain
D
in
H
(for
simplicity), and assume that 0
∈ ∂D
. Fixing a point
y ∈ ∂D
, we perform the
percolation exploration as before. Then the resulting path would look exactly
the same as if we performed percolation on
H
(with black boundary on
R
<0
and
white boundary on
R
>0
), up to the point we hit
∂D \ ∂H
. In other words,
γ
doesn’t feel the boundary conditions until it hits the boundary. This is known
as locality.
It should then be true that the corresponding
SLE
κ
should have an analogous
property. To be precise, we want to find a value of
κ
so that the following is
true: If
γ
is an
SLE
κ
in
H
from 0 to
∞
, run up until it first hits
∂D \∂H
, then
ψ
(
γ
) is a (stopped)
SLE
κ
in
H
from 0 to
∞
where
ψ
:
D → H
is a conformal
transformation with
ψ
(0) = 0,
ψ
(
y
) =
∞
. This is the locality property. We will
show that locality holds precisely for κ = 6.
Suppose that (A
t
) ∈ A has Loewner driving function U
t
. We define
˜
A
t
= ψ(A
t
).
Then
˜
A
t
is a family of compact
H
-hulls that are non-decreasing, locally growing,
and
A
0
=
∅
. However, in general, this is not going to be parametrized by capacity.
On the second example sheet, we show that this has half plane capacity
˜a(t) = hcap(
˜
A
t
) =
Z
t
0
γ(ψ
0
t
(U
s
))
2
ds,
which should be somewhat believable, given that hcap(rA) = r
2
hcap(A).
˜
A
t
has a “driving function”
˜
U
t
given by
˜
U
t
= ψ
t
(U
t
), ψ
t
= ˜g
t
◦ ψ ◦ g
−1
t
, g
t
= g
A
t
.
We then have
∂
t
˜g
t
(z) =
∂
t
˜a
t
˜g
t
(z) −
˜
U
t
, ˜g
0
(z),
To see this, simply recall that if A
t
is γ([0, t]) for a curve γ, then U
t
= g
t
(γ(t)).
To understand
˜
U
t
, it is convenient to know something about ψ
t
:
Proposition. The maps (ψ
t
) satisfy
∂
t
ψ
t
(z) = 2
(ψ
0
t
(U
t
))
2
ψ
t
(z) −ψ
t
(U
t
)
−
ψ
0
t
(U
t
)
z − U
t
.
In particular, at z = U
t
, we have
∂
t
ψ
t
(U
t
) = lim
z→U
t
∂
t
ψ
t
(z) = −3ψ
00
t
(U
t
).
Proof. These are essentially basic calculus computations.
To get Loewner’s equation in the right shape, we perform a time change
σ(t) = inf
u ≥ 0 :
Z
u
0
(ψ
0
s
(U
s
))
2
ds = t
.
Then we have
∂
t
˜g
σ( t)
(z) =
2
˜g
σ(t)
−
˜
U
σ(t)
, ˜g
0
(z) = z.
It then remains to try to understand d
˜
U
σ(t)
. Note that so far what we said
works for general
U
t
. If we put
U
t
=
√
κB
t
, where
B
is a standard Brownian
motion, then Itˆo’s formula tells us before the time change, we have
d
˜
U
t
= dψ
t
(U
t
)
=
∂
t
ψ
t
(U
t
) +
κ
2
ψ
00
t
(U
t
)
dt +
√
κψ
0
t
(U
t
) dB
t
=
κ − 6
2
ψ
00
t
(U
t
) dt +
√
κψ
0
t
(U
t
) dB
t
.
After the time change, we get
d
˜
U
σ(t)
=
κ − 6
2
ψ
00
σ(t)
(U
σ(t)
)
ψ
0
σ(t)
(U
σ(t)
)
2
dt +
√
κ d
˜
B
t
,
where
˜
B
t
=
Z
σ( t)
0
ψ
0
s
(U
s
) dB
s
is a standard Brownian motion by the L´evy characterization and the definition
of σ(t).
The point is that when κ = 6, the drift term goes away, and so
˜
U
σ(t)
=
√
6
˜
B
t
.
So (
˜
A
σ(t)
) is an SLE
6
. Thus, we have proved that
Theorem.
If
γ
is an
SLE
κ
, then
ψ
(
γ
) is an
SLE
κ
up until hitting
ψ
(
∂D \∂H
)
if and only if κ = 6.
So κ = 6 is the only possible SLE
κ
which could be the limit of percolation.