An Introduction to the Schramm-Loewner Evolution
Tom Alberts
Cou(r)an(t) Institute
March 14, 2008
Outline
• Lattice models whose time evolution is not Markovian.
• Conformal invariance of their scaling limits.
• Their common Domain Markov Property.
• The SLE scaling limit
Simple Random Walk
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Simple Random Walk
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Simple Random Walk
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Simple Random Walk
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Simple Random Walk
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Simple Random Walk
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Simple Random Walk
• Here we think of simple random walk as a law on paths
(without a time parameterization) in some grid domain
with mesh size δ. All paths start at some point z and stop
when they hit the boundary of the domain.
• Refer to this law as SRWδ (z, D), where D is the domain.
• Important question: does the law SRWδ (z, D) converge to
something as δ ↓ 0?
• In this case the limiting law is the Brownian law on paths.
• Refer to this law as µ(z, D).
Conformal Invariance of Brownian Motion
• Suppose f : D → D′ is a conformal map, with D and D′
simply connected. Let Bt be a complex Brownian motion.
• Then f (Bt), modulo a time-change, is also a Brownian
motion.
• Easily proved by Ito’s Lemma.
• Consequently the measure µ(z, D) is conformally
invariant:
f ◦ µ(z, D) = µ (f (z), f (D))
Conformal Invariance of a Lattice Model
conformally map
approximate the region
Models with Conformal Invariance but no Markov Property
• There are very simple lattice models that have (or are
conjectured to have) conformally invariant scaling limits,
but whose “time evolution” is not Markovian.
• Schramm-Loewner Evolution was introduced by [Sch00]
as a possible candidate for the scaling limit.
• Has proven to be extremely successful, and is proven
to be the scaling limit of at least 4 different models. Is
conjectured to be the scaling limit of many more.
• Here I describe two different lattice models that have
an SLE scaling limit: loop erased random walk and the
percolation exploration process.
Loop Erased Random Walk
• Very simple model to understand.
• Derived from simple random walk by a loop erasing
operation.
• First sample a simple random walk path, and then walk
through the path and chronologically erase the loops in
the order they are encountered.
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
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Loop Erased Random Walk
• The loop erasing procedure induces a new law on paths
from z to D, on a grid with mesh size δ. Denote this new
law by LERWδ (z, D).
• If a scaling limit exists, then one expects it to be
conformally invariant, i.e.
f ◦ LERW0+ (z, D) = LERW0+ (f (z), f (D))
• In fact, the scaling limit should correspond to “loop erased
Brownian motion”.
• However, there is no canonical way to loop erase a
Brownian path, so the scaling limit can’t be as simple as
that.
Loop Erased Random Walk
• The time evolution of loop erased random walk is not
Markovian.
• The curve can’t intersect itself, so the future path is
always highly correlated with its past.
• Still, there is a Domain Markov Property.
Domain Markov Property of LERW
• First we fix a boundary point y ∈ ∂D, and consider the
law of LERW conditioned to exit D at y.
• Call this law LERWδ (z, y, D)
y
z
• For LERWδ (z, y, D), it makes sense to reverse the
direction of the path so that it goes from y to z.
Domain Markov Property of LERW
• Now suppose we’ve observed the first step of the loop
erased random walk.
• What is the law of the remaining path?
y
y1
z
• Expect it to be, and
LERWδ (z, y1, D\[y, y1]).
it
turns
out
that
it
is,
Domain Markov Property of LERW
• Can easily generalize to having observed the first k steps
of the loop erased walk.
• Law of remaining path is LERWδ (z, yk , D\[y, yk ]).
y
yk
z
Domain Markov Property of LERW
• Constrast the Domain Markov Property with the one for
SRW.
• For SRW, having observed part of the path, the remainder
has the law of a SRW in the same domain D, started from
where the observed path left off.
• For LERW, having observed part of the path, the
remainder has the law of a LERW in the domain
D\observed path, started from where the observed path
left off.
• Thus the LERW process is somehow “Markovian in the
evolution of the domain”.
Percolation Exploration Process
Percolation Exploration Process
Percolation Exploration Process
Percolation Exploration Process
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Percolation Exploration Process
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Percolation Exploration Process
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Percolation Exploration Process
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Percolation Exploration Process
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Percolation Exploration Process
• Again the curve can’t intersect itself, so the future path is
highly correlated with the past.
• Evolution of the curve is not Markovian in time.
• There is still a Domain Markov Property in the model.
• It is still expected that the scaling limit is conformally
invariant (if it exists).
Domain Markov Property of Percolation Exploration
Process
• Suppose we’ve observed the exploration curve after a
certain number of steps.
Domain Markov Property of Percolation Exploration
Process
• Suppose we’ve observed the exploration curve after a
certain number of steps.
Domain Markov Property of Percolation Exploration
Process
• Suppose we’ve observed the exploration curve after a
certain number of steps.
Domain Markov Property of Percolation Exploration
Process
• The law of the remaining path is that of a percolation
exploration process in the original domain less the slit
created by the observed curve.
• This is the same Domain Markov Property that LERW
has.
Possible Scaling Limits
• If a scaling limit exists for either LERW or the percolation
exploration process, it should have:
* Conformal Invariance
* Domain Markov Property
• If one assumes that there is a law on paths (modulo
time parameterization) with these two properties, then
the Loewner equation from complex analysis plus some
probability theory determines exactly what the law has to
be.
Possible Scaling Limits
• Here I describe the law on paths from one boundary
point of the domain to another (as for the percolation
exploration process). This law is called chordal SLE.
• By the conformal invariance property, it’s enough to
describe the law for a canonical simply connected domain
and any two boundary points.
• The canonical choice is the upper half-plane H, with
curves starting at 0 and going to ∞.
• There is also a law on paths going from a boundary point
to an interior point (as for LERW). This is called radial
SLE.
• Here the canonical choice is paths in the unit circle going
from 1 to 0.
Loewner Equation
• Consider a non self crossing curve γ : [0, ∞) → H such
that γ(0) = 0, γ(∞) = ∞.
• Then ∀t ≥ 0, H\γ[0, t] is a simply connected domain.
gt
γ([0, t])
0
Ut = gt (γ(t))
• Riemann Mapping Theorem says there exists a conformal
map gt : H\γ[0, t] → H.
• Loewner Equation describes how the maps gt evolve as
the curve γ[0, t] grows.
Loewner Equation
• Consider a non self crossing curve γ : [0, ∞) → H such
that γ(0) = 0, γ(∞) = ∞.
• Then ∀t ≥ 0, H\γ[0, t] is a simply connected domain.
gt
γ([0, t])
0
Ut = gt (γ(t))
2
,
∂tgt(z) =
gt(z) − Ut
g0(z) = z
Basic Idea of Loewner Equation
• Supposing that we know gt, can we use it to find gt+dt?.
γ([t, t + dt])
gt
γ([0, t])
gt (γ[t, t + dt])
0
Ut
ht,dt : H\gt (γ[t, t + dt]) → H
gt+dt
Ut+dt
• Can compute the map ht,dt explicitly.
Basic Idea of Loewner Equation
w → (w − Ut )2
√
2dt
Ut
−2dt
0
w→
p
ht,dt(w) = Ut+dt + (w − Ut)2 + 2dt
2 dt
≈ w+
w − Ut
gt+dt(z) = ht,dt (gt(z))
2 dt
≈ gt(z) +
gt(z) − Ut
√
w + 2dt
Basic Idea of Loewner Equation
w → (w − Ut )2
√
2dt
Ut
−2dt
0
w→
2
,
∂tgt(z) =
gt(z) − Ut
g0(z) = z
√
w + 2dt
Technicalities of Loewner Equation
• First note that gt still has three real degrees of freedom.
• Choose two as follows: gt(∞) = ∞, gt′ (∞) = 1.
• Laurent expansion at ∞ must look like
b1 b2
gt(z) = z + a0 + + 2 + . . .
z z
• Set a0 = 0.
• All other coefficients are now fixed.
• Two important quantities: Ut = gt(γ(t)) and b1.
• The b1 coefficient (which is a function of t) is called the
half-plane capacity of γ[0, t], and denoted by a (γ[0, t]).
Technicalities of Loewner Equation
• First note that gt still has three real degrees of freedom.
• Choose two as follows: gt(∞) = ∞, gt′ (∞) = 1.
• Laurent expansion at ∞ must look like
a (γ[0, t]) b2
+ 2 + ...
gt(z) = z +
z
z
• Set a0 = 0.
• All other coefficients are now fixed.
• Two important quantities: Ut = gt(γ(t)) and b1.
• The b1 coefficient (which is a function of t) is called the
half-plane capacity of γ[0, t], and denoted by a (γ[0, t]).
Technicalities of Loewner Equation
• It is easy to check that a (γ[0, t]) is additive, i.e.
a (γ[0, t + s]) = a (γ[0, t]) + a (gt (γ[t, t + s]))
γ([t, t + s])
gt
γ([0, t])
gt (γ[t, t + s])
0
Ut = gt (γ(t))
• Can also show that a (γ[0, t]) is continuous and increasing,
so can reparameterize the curve so that a (γ[0, t]) = 2t.
2t
gt(z) = z + + . . .
z
Using the Loewner Equation to Produce Curves
• So given the curve γ, the maps gt must satisfy
2
,
∂tgt(z) =
gt(z) − Ut
g0(z) = z
where Ut = gt(γ(t)).
• Can turn the situation around.
Given the driving
function Ut : [0, ∞) → R, the ODE can be solved to
determine gt. The gt then determine γ[0, t].
• It’s not a priori true that any driving function Ut will output
a curve. A sufficient condition is that Ut be Holder
continuous with exponent greater than 1/2.
Candidates for the SLE Driving Function
• To produce random curves, Ut obviously has to be
random. Should also be continuous.
• Inputting a random Ut induces a law on curves in the
upper half plane.
• Claim: If this law satisfies the conformal invariance and
Domain Markov properties, then it must be that Ut = σBt
for some σ > 0, where Bt is a standard Brownian motion.
Candidates for the SLE Driving Function
• Domain Markov: Given the curve γ[0, t], the law of
γ[t, ∞) is chordal SLE in H\γ[0, t] from γ(t) to ∞.
• Conformal Invariance: The chordal SLE law in H\γ[0, t]
from γ(t) to ∞ is the pullback of the chordal SLE law in H
from Ut to ∞.
gt
γ([0, t])
0
γ ∗ (s) = gt (γ(t + s))
Ut = gt (γ(t))
• Hence γ ∗(s) has the law of a chordal SLE curve in H from
Ut to ∞ that is independent of γ[0, t].
• But γ ∗(s) is determined by the driving function Ut+s −
Ut, s ≥ 0, and γ[0, t] by the driving function Us, 0 ≤ s ≤ t.
Candidates for the SLE Driving Function
• Then γ ∗(s) independent of γ[0, t] means that the driving
function after time t should be independent of the driving
function up to time t, i.e. Ut is a Markov process.
• Moreover, γ ∗[0, t] is identical in law to γ[0, t], so Us, 0 ≤
s ≤ t, should have the same law as Ut+s − Ut, 0 ≤ s ≤ t.
gt
γ([0, t])
0
γ ∗ (s) = gt (γ(t + s))
Ut = gt (γ(t))
• This forces the increments of Ut to be i.i.d., and writing
dUt = b dt + σdBt
this is only possible if b and σ are constants.
Candidates for the SLE Driving Function
• Hence conformal invariance and the Domain Markov
property force Ut to be Brownian motion with drift.
• One extra consideration: the law of the SLE curve should
be invariant under reflections about the imaginary axis.
• This means the law of Ut should be invariant under
negation.
• Hence b = 0, leaving dUt = σdBt.
√
• Schramm writes Ut = κBt.
SLE
• Formal definition of chordal SLE(κ): the solution to the
ODE
2
√
∂tgt(z) =
,
gt(z) − κBt
g0(z) = z
where Bt is a standard one-dimensional Brownian motion
with B0 = 0. The curve γ corresponding to the maps gt is
called the chordal SLE(κ) curve.
• Since Bt is not Holder-1/2 continuous, it’s not a priori true
that a curve is produced for SLE(κ). But [RS05] proves
that it is true.
The Dependence on κ
• κ looks like an innocent parameter, but it actually plays a
very important role.
• κ = 0:
2
,
∂tgt(z) =
gt(z)
gt(z) =
√
√
2 t
0
g0(z) = z
z 2 + 4t
The Dependence on κ
• As soon as κ > 0 the curve is immediately fractal. As κ
increases it becomes more fractal.
• [Bef07] proves that the Hausdorff
dimension of the curve
is almost surely min 1 + κ8 , 2 .
• There are also phase transitions at κ = 4 and κ = 8.
The Dependence on κ
• For κ ≤ 4, the curve is almost surely simple (i.e. doesn’t
touch itself).
• Moreover, it doesn’t intersect the real line, i.e. γ ∩ R = {0}
• For a fixed z ∈ H, almost surely the curve does not hit z.
SLE(2)
The Dependence on κ
• For 4 < κ < 8, the curve touches itself (but does not cross
itself).
• The curve intersects part of the real line, i.e. γ ∩ R ( R.
• For a fixed z ∈ H, almost surely the curve does not hit z,
but does make a loop around z.
SLE(6)
The Dependence on κ
• For κ ≥ 8, the curve is space filling.
• Hits every point on the real line, i.e. γ ∩ R = R.
• For a fixed z ∈ H, almost surely the curve does hit z.
SLE(κ > 8)
What κ Correspond to LERW and Percolation?
• For LERW expect κ ≤ 4, for percolation expect κ > 4.
• Turns out that LERW corresponds to κ = 2.
• Percolation corresponds to κ = 6.
• To determine this, one has to compute some specific
probability or statistic for the lattice model, then compute
the same thing for SLE(κ).
• There’s only one value of κ for which these two quantities
agree.
Models with an SLE Scaling Limit
• κ = 2: Loop-Erased Walk
• κ = 8/3: Self-Avoiding Walk
• κ = 3: Ising Interface
• κ = 4: Harmonic Explorer and the Gaussian Free Field
interface
• κ = 6: Percolation Exploration Process
• κ = 8: Uniform Spanning Tree
References
[AS07] Tom Alberts and Scott Sheffield. Hausdorff dimension
of the sle curve intersected with the real line.
arXiv:0711.4070 [math.PR], 2007.
[Bef07] Vincent Beffara. The dimension of the SLE curves. To
appear in Ann. Prob., 2007.
[RS05] Steffen Rohde and Oded Schramm. Basic properties
of SLE. Ann. of Math. (2), 161(2):883–924, 2005.
[Sch00] Oded Schramm. Scaling limits of loop-erased random
walks and uniform spanning trees. Israel J. Math.,
118:221–288, 2000.
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