2Loewner's theorem
III Schramm--Loewner Evolutions
2.2 Schramm–Loewner evolution
We now take
U
t
to be a random process adapted to
F
t
=
σ
(
U
s
:
s ≤ t
). We can
then define
g
t
(
z
) as the solution to the Loewner equation for each
z
, and define
A
t
accordingly. We then obtain a random family A
t
in A.
Definition
(Conformal Markov property)
.
We say that (
A
t
) satisfy the confor-
mal Markov property if
(i) Given F
t
, (g
t
(A
t+s
) − U
t
)
s≥0
d
= (A
s
)
s≥0
.
(ii) Scale-invariance: (rA
t/r
2
)
t≥0
d
= (A
t
).
Here (i) is the Markov property, and the second part is the conformal part,
since the only conformal maps H → H that fix 0 and ∞ are the rescalings.
Schramm was interested in such processes because they provide us with
potential scaling limits of random processes, such as self-avoiding walk or perco-
lation. If we start with a process that has the Markov property, then the scaling
limit, if exists, ought to be scale invariant, and thus have the conformal Markov
property. Schramm classified the list of all possible such scaling limits:
Theorem
(Schramm)
.
If (
A
t
) satisfy the conformal Markov property, then there
exists κ ≥ 0 so that U
t
=
√
κB
t
, where B is a standard Brownian motion.
Proof.
The first property is exactly the same thing as saying that given
F
t
, we
have
(U
t+s
− U
t
)
s≥0
d
= (U
s
).
So
U
t
is a continuous process with stationary, independent increments. This
implies there exists κ ≥ 0 and a ∈ R such that
U
t
=
√
κB
t
+ at,
where B is a standard Brownian motion. Then the second part says
(rU
t/r
2
)
t≥0
d
= (U
t
)
t≥0
.
So U satisfies Brownian scaling. Plugging this in, we know
r
√
κB
t/r
2
+ at/r
d
= r
√
κB
t
+ at.
Since r
√
κB
t/r
2
d
= r
√
κB
t
, we must have a = 0.
This finally gives us the definition of SLE.
Definition
(Schramm–Loewner evolution)
.
For
κ >
0,
SLE
κ
is the random
family of hulls encoded by
U
t
=
√
κB
t
, where
B
is a standard Brownian motion.
When κ = 0, we have U
t
= 0. This corresponds to the curve γ(t) = 2
√
ti.
Usually, when talking about SLE
κ
, we are instead thinking about a curve.
Theorem
(Rhode–Schramm, 2005)
.
If (
A
t
) is an
SLE
κ
with flow
g
t
and driving
function
U
t
, then
g
−1
t
:
H → A
t
extends to a map on
¯
H
for all
t ≥
0 almost surely.
Moreover, if we set
γ
(
t
) =
g
−1
t
(
U
t
), then
H \ A
t
is the unbounded component of
H \ γ([0, t]).
We will not prove this. From now on,
SLE
κ
will often refer to this curve
instead.
In the remainder of the course, we will first discuss some properties of
SLE
κ
,
and then “prove” that the scaling limit of certain objects are
SLE
κ
’s. We will not
be providing anything near a full proof of the identification of the scaling limit.
Instead, we will look at certain random processes, and based on the properties of
the random process, we deduce that the scaling limit “ought to” have a certain
analogous property. We will then show that for a very particular choice of
κ
,
the curve
SLE
κ
has that property. We can then deduce that the scaling limit is
SLE
κ
for this κ.
What we will do is to prove properly that the
SLE
κ
does indeed have
the desired property. What we will not do is to prove that the scaling limit
exists, satisfies the conformal Markov property and actually satisfies the desired
properties (though the latter two should be clear once we can prove the scaling
limit exists).