Definitions and background
Outline
Almost sure multifractal spectrum of SLE
Ewain Gwynne
(Joint with Jason Miller and Xin Sun)
1
Definitions and background
2
Upper bound
3
Lower bound
4
A few details of the proof
5
Conclusion
Massachusetts Institute of Technology
Conformally invariant scaling limits, University of Cambridge
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
1 / 59
Ewain Gwynne (MIT)
Definitions and background
Multifracal spectrum
Let
e s (D) :=
Θ
The multifractal spectrum of D is a means of quantifying the
behavior of |φ0 | (resp. |(φ−1 )0 |) near ∂D (resp. ∂D).
Almost sure multifractal spectrum of SLE
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Multifractal spectrum
Let D ⊂ C be a simply connected domain (e.g. a complementary
connected component of an SLEκ curve). Let φ : D → D be a
conformal map.
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
Definitions and background
x ∈ ∂D : lim
→0
log |φ0 ((1 − )x)|
=s .
− log e s (D)) ⊂ ∂D.
Let Θs (D) := φ(Θ
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
4 / 59
Definitions and background
Definitions and background
Multifractal spectrum
Multifractal spectrum
φ
e s (D)
The multifractal spectrum of D is the two functions s 7→ dimH Θ
and s 7→ dimH Θs (D).
e s (D) = Θs (D) = ∅ for s ∈
We have Θ
/ [−1, 1], so this is only of
D
interest for s ∈ [−1, 1].
Related to, e.g., the harmonic measure spectrum of D, the integral
means spectrum of D, the Hölder regularity of φ, and the Hausdorff
dimension of ∂D.
(1 − )x
x
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Almost sure multifractal spectrum of SLE
5 / 59
Ewain Gwynne (MIT)
Definitions and background
Related results
6 / 59
Multifractal spectrum
Hausdorff dimension computed by Beffara (2008).
Hölder exponent computed by Lawler and Viklund (2011) building on
works by Rohde and Schramm (2005) and Lind (2008).
Non-rigorous predictions for the multifractal spectrum by Duplantier
as early as 2000.
Lead Duplantier to conjecture SLE duality, the statement that the
outer boundary of an SLEκ for κ > 4 locally looks like an SLE16/κ
(rigorously established in works by Dubedát, Zhan, Miller-Sheffield)
Almost sure multifractal spectrum of SLE
Theorem: (Gwynne, Miller, Sun) Let κ > 0 and let η be an SLEκ in a
smoothly bounded domain D ⊂ C. Let
√ p
√ p
4κ − 2 2 κ(2 + κ)(8 + κ)
4κ + 2 2 κ(2 + κ)(8 + κ)
s− =
, s+ =
.
(4 + κ)2
(4 + κ)2
Let s ∈ [s− , s+ ]. Almost surely, for each t > 0 and each complementary
connected V of η([0, t]), we have
Lawler and Viklund (2012) computed the multifractal spectrum at the
tip of SLE.
Beliaev and Smirnov (2009) computed the average integral means
spectrum of SLE.
Alberts, Binder, and Viklund (2015) computed a dimension spectrum
for points where SLE hits the boundary.
More in later talks today.
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
Definitions and background
7 / 59
(4 + κ)2 s 2
8κ(1 + s)
8κ(1 + s − s 2 ) − 16s 2 − κ2 s 2
dimH Θs (V ) =
.
8κ(1 − s 2 )
e s (V ) = 1 −
dimH Θ
e s (V ) = Θs (V ) = ∅.
For s ∈
/ [s− , s+ ], a.s. Θ
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Almost sure multifractal spectrum of SLE
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Definitions and background
Definitions and background
Multifractal spectrum
Multifractal spectrum
1.2
1.0
s-
0.8
ξ(s)
s+
˜
ξ (s)
Agrees with predictions of Duplantier.
Invariant under replacing κ with 16/κ (SLE duality).
s 7→ ξ(s) is maximized at s = κ/4, where is equals 1 + κ/8.
0.6
-1.0
Ewain Gwynne (MIT)
0.4
κ
0.2
4
0.0
-0.5
0.5
This yields an alternative proof that dimH η = 1 + κ/8 a.s. for
κ ∈ (0, 4].
1.0
Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Definitions and background
Almost sure multifractal spectrum of SLE
10 / 59
Definitions and background
Integral means spectrum
Integral means spectrum
The integral means spectrum of D is the function IMSD : R → R
defined by
R
log ∂B1− (0) |φ0 (z)|a dz
,
IMSD (a) = lim sup
− log →0
Average integral means spectrum of SLE computed by Beliaev-Smirnov
(2009):
R 2π
log 0 E|(ft−1 )0 (re iθ )|a dθ
.
lim sup
− log(r − 1)
+
r →1
where φ : D → D is a conformal map.
Related to several conjectures in complex analysis.
Usually hard to compute for deterministic fractals, but can be easier
for random fractals.
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Definitions and background
Definitions and background
Integral means spectrum
Integral means spectrum
3.0
We obtain the a.s. bulk integral means spectrum of SLE (which is
defined in the same way as the ordinary integral means spectrum, but
with small neighborhoods of the tip and starting point of η removed).
2.5
1.5
1.0
0.5
-6
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Almost sure multifractal spectrum of SLE
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2.0
a-
Corollary: Let κ > 0 and let η be an SLEκ in a smoothly bounded
domain D ⊂ C. Almost surely, for each t > 0, each a ∈ R, and each
complementary connected V of η([0, t]), we have


a < a−
−1 + s− a,

√
2
bulk
IMSV (a) = −a + (4+κ)(4+κ− (4+κ) −8aκ) ,
a ∈ [a− , a+ ]
4κ


−1 + s a,
a > a+ .
+
Ewain Gwynne (MIT)
Upper bound
-4
-2
0
a+
2
Almost sure multifractal spectrum of SLE
4
6
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Upper bound
Outline
Setup
1
Definitions and background
2
Upper bound
To establish an upper bound for the Hausdorff dimension of the sets
e s (D), we need to estimate the probability that a point is
Θs (D) and Θ
contained in these sets.
3
Lower bound
By SLE duality it suffices to consider κ ≤ 4.
4
A few details of the proof
5
Conclusion
Ewain Gwynne (MIT)
By a change of coordinates, we can assume that η is a chordal SLEκ
from −i to i in D. Let Dη be the right complementary connected
component of η.
Also let Ψ : Dη → D be the conformal map which fixes −i, i, and 1.
Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Upper bound
Upper bound
Setup
Setup
Ψ
i
i
To establish an upper bound for the Hausdorff dimension of the sets
e s (D), we need to estimate the probability that a point is
Θs (D) and Θ
contained in these sets.
By SLE duality it suffices to consider κ ≤ 4.
Dη
1
By a change of coordinates, we can assume that η is a chordal SLEκ
from −i to i in D. Let Dη be the right complementary connected
component of η.
1
Also let Ψ : Dη → D be the conformal map which fixes −i, i, and 1.
Dη will be convenient for the two-point estimate because we can grow
the curve from both directions.
−i
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−i
Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Upper bound
Reverse SLE
Reverse SLE/GFF
coupling
Pointwise derivative
estimates for finite
time inverse maps
Reverse SLEκ is obtained by solving the reverse Loewner equation
Change of
variables
Estimates for the area of
the set where forward
finite time maps have
given derivative behavior
ġt (z) = −
Markov property Pointwise derivative
estimates for time
and regularity
infinity map
conditions
Usual
argument
Upper bound for
dimension of
Ewain Gwynne (MIT)
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Upper bound
Upper bound
Reverse SLE
martingales
Almost sure multifractal spectrum of SLE
2
.
gt (z) − Ut
The solution is a family of conformal maps gt : H → H \ Kt , for (Kt )
hulls in H.
√
If Ut = κBt , then gt − Ut has the same law as the time t centered
Loewner map of a forward SLEκ .
Usual
argument
Upper bound for
dimension of
Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Upper bound
Upper bound
Reverse SLE
One point estimate for inverse maps
If z ∈ H and ρ ∈ R, then
gt
2 /8κ
Mt = |gt0 (z)|(8+2κ−ρ)ρ/8κ (Im gt (z))−ρ
|gt (z) −
√
κBt |ρ/κ
is a martingale.
Introduced by Lawler (2009).
Reverse analogue of the Schramm-Wilson martingales for forward
SLE.
Re-weighting by Mt gives a reverse SLEκ (ρ) with a force point at z.
Call this reweighted law Pz∗ .
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Upper bound
One point estimate for inverse maps
If Im z = and z is not too close to 0 or ∞, then
P |gt0 (z)| ≈ −s ≈ α Pz∗ |gt0 (z)| ≈ −s
To show Pz∗ (|gt0 (z)| ≈ −s ) 1 we use a coupling with a Gaussian
free field.
where ≈ means −s+u ≤ |gt0 (z)| ≤ −s−u for u > 0 small but fixed.
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Almost sure multifractal spectrum of SLE
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Upper bound
One point estimate for inverse maps
z
We claim that if we take ρ = (4+κ)s
1+s , then P∗
Given this we obtain P (|gt0 (z)| ≈ −s ) ≈ α .
Almost sure multifractal spectrum of SLE
(|gt0 (z)|
≈
−s )
1.
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By a theorem of Sheffield (2011) we can find random distributions h
d
and ht (GFF’s plus harmonic functions) s.t. h ◦ gt + Q log |gt0 | = ht
√
√
under P∗z , where Q = 2/ κ + κ/2.
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Almost sure multifractal spectrum of SLE
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Upper bound
Upper bound
One point estimate for inverse maps
One point estimate for inverse maps
To show Pz∗ (|gt0 (z)| ≈ −s ) 1 we use a coupling with a Gaussian
free field.
h ◦ gt + Q log |gt0 |
gt
By a theorem of Sheffield (2011) we can find random distributions h
h
d
and ht (GFF’s plus harmonic functions) s.t. h ◦ gt + Q log |gt0 | = ht
√
√
under P∗z , where Q = 2/ κ + κ/2.
By estimating the circle average processes for h and ht , we get that
|gt0 (z)| ≈ −s with high probability under P∗z .
This leads to
P |gt0 (z)| ≈ −s ≈ α(s) .
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Upper bound
Upper bound
Using stochastic calculus and a symmetry between forward and
reverse SLEκ (ρ) due to Duplantier, Miller, and Sheffield (2014), we
can also add extra regularity conditions to our lower bound.
This is the most technical part of the one-point estimate.
Our derivative estimates allow us to estimate the expected number of
e s (H \ η([0, t])).
-balls needed to cover Θ
This gives an upper bound for the Hausdorff dimension of
e s (H \ η([0, t])) and the integral means spectrum of H \ η([0, t]).
Θ
Basic complex analysis allows us to transfer these upper bounds to
Dη .
This is also the main reason why we can’t just cite other similar
results in the literature (e.g. Rohde-Schramm, Beliaev-Smirnov).
Almost sure multifractal spectrum of SLE
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Upper bound
One point estimate for inverse maps
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Upper bound
Upper bound
Area estimate
One point estimate for the forward maps
Ψ
i
By a change of variables and the Koebe quarter theorem
(|ft0 (z)| dist(z, η) Im ft (z)), we can estimate the area of the set of
z ∈ H with |ft0 (z)| ≈ s and dist(z, η([0, t])) ≈ 1−s .
i
z
Dη
We then transfer this to an estimate for the area of the set As of
z ∈ Dη for which |Ψ0 (z)| ≈ s and dist(z, η) ≈ 1−s .
1
−i
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
Ewain Gwynne (MIT)
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1
−i
Almost sure multifractal spectrum of SLE
Upper bound
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Lower bound
One point estimate for the forward maps
Outline
Using the Markov property, one can argue that P (z ∈ As ) does not
depend too strongly on z.
This takes us from area estimates to pointwise estimates:
P |Ψ0 (z)| ≈ s , dist(z, η) ≈ 1−s ≈ γ(s) .
1
Definitions and background
2
Upper bound
3
Lower bound
4
A few details of the proof
5
Conclusion
This estimate leads to an upper bound for dimH Θs (Dη ).
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Lower bound
Lower bound
Lower bound
Lower bound
We prove a lower bound for dimH Θs (Dη ) first.
This will allow us to construct a Frostman measure on a self-similar subset
of Θs (Dη ) (the “perfect points”), i.e. a positive finite measure satisfying
Z Z
1
dν(z)dν(w ) < ∞
|z − w |α
Let q := s/(1 − s).
We define nested events En (z) for z ∈ D such that
T∞
s
n=1 En (z) ⊂ {z ∗∈ Θ (Dη )}.
P (En (z)) ≈ e −βγ (q)n .
P(En (z)∩En (w ))
γ ∗ (q)+o|z−w | (1)
.
P(En (z))P(En (w )) ≤ |z − w |
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
for given α < ξ(s).
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Almost sure multifractal spectrum of SLE
Lower bound
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Lower bound
Lower bound
Lower bound
φβ
i
Let η be the time reversal of η, which is an SLEκ from −i to i.
i
η̄
We will look at the behavior of η and η at the first time they hit
Be −β (z).
1
1
η
−i
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
−i
Almost sure multifractal spectrum of SLE
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Lower bound
Lower bound
Lower bound
Lower bound
i
ψ0,j
i
Now we iterate this.
Let η0,1 = η. Inductively let η0,j+1 be the curve obtained by running
η0,j and η 0,j up to the first time they Be −β (0), then applying the map
ψ0,j which takes the complement of the two sides of the curve to D,
with 0 fixed.
d
η̄0,j
η̄0,j+1
η0,j
η0,j+1
−i
−i
Then we have η0,j = η, modulo perturbations of the endpoints (which
we can deal with by growing out a little more of the curve).
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Lower bound
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Lower bound
Lower bound
Lower bound
Let φ0,j be defined in the same manner as φβ above, but with η0,j in
place of η.
Let E0,j be the event that |φ0j (0)| ≈ e −βq (plus a bunch of regularity
conditions).
T
Let En (0) = nj=1 E0,j .
Define En (z) for z ∈ D by first mapping z to 0.
Define the perfect points to be the (approximately) the set of z ∈ D
for which En (z) occurs.
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
Almost sure multifractal spectrum of SLE
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Our events are set up so that
T∞
n=1 En (z)
⊂ Θs (Dη ).
The probability that |φ0β (0)| ≈ e −βq is of the same order as the
probability that |Ψ0 (0)| ≈ e −βq and dist(z, η) ≈ e −β .
We know the latter probability is e −βγ
(with = e −β/(1−s) ).
Hence P(En (z)) ≈
Ewain Gwynne (MIT)
∗ (q)
by the one-point estimate
∗
e −nβγ (q) .
Almost sure multifractal spectrum of SLE
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Lower bound
Lower bound
Lower bound
Lower bound
ψz,k
Consider two points z and w with |z − w | ≈ e −kβ .
η̄z,k
We need to estimate P(En (z) ∩ En (w )) for n ≥ k.
w
z
The points 0 = ψz,k (z) and ψz,k (w ) are at constant order distance
apart.
ηz,k
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Lower bound
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Lower bound
Lower bound
Flow lines
We can couple η with a GFF h on D in such a way that η is the “flow
line” of h started from −i (in the sense of Miller and Sheffield’s
Imaginary Geometry papers).
We would like to say that the behaviors of the curve near the two
points are approximately independent.
However, we are interested in the derivative of a certain conformal
map, which may depend on the whole curve.
At each stage in the construction of En (z), we add auxiliary flow lines
±
ηz,j
for h started from the tip of the part of η we have grown so far.
To get around this we need to localize.
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Lower bound
Lower bound
Flow lines
Flow lines
The auxiliary flow lines form “pockets” with the property that the
intersection of η with each pocket is conditionally independent of
what happens outside the pocket, given the pocket.
η̄z,2
η̄
+
ηz,2
+
ηz,1
We re-define the curves ηz,j so that they only depend on the part of
the curve inside the j − 1th pocket around z.
−
ηz,2
−
ηz,1
ηz,2
η
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Lower bound
Almost sure multifractal spectrum of SLE
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Lower bound
Flow lines
Lower bound
This leads to an estimate for
+
ηw,k+1
η̄
−
ηw,k+1
w
z
+
ηz,k+1
+
ηz,k
η
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
in terms of |z − w |.
Can use the same estimates (and some relatively minor tricks) to get
e s (Dη ) and IMSbulk (Dη ).
a lower bound for dimH Θ
−
ηz,k+1
−
ηz,k
P(En (z)∩En (w ))
P(En (z))P(En (w ))
Once we have such an estimate, we get a lower bound for
dimH Θs (Dη ) via the usual (Frostman measure) argument.
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Almost sure multifractal spectrum of SLE
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A few details of the proof
A few details of the proof
Outline
Reverse continuity conditions
1
Definitions and background
2
Upper bound
3
Lower bound
4
A few details of the proof
5
Conclusion
Throughout this talk, all of our events have involved “regularity
conditions”. The most important (but by no means the only) such
regularity conditions are the following.
Let A ⊂ D be a closed set. Let D be a connected component of D \ A
and let f : D → D be a conformal map. Let µ : (0, ∞) → (0, ∞) be
an increasing function.
G(f , µ): for each δ > 0 and each x, y ∈ ∂D ∩ ∂D with |x − y | ≥ δ,
we have |f (x) − f (y )| ≥ µ(δ).
G 0 (A, µ): for each δ > 0, A lies at distance at least µ(δ) from
∂D \ (A ∩ ∂D).
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
A few details of the proof
Almost sure multifractal spectrum of SLE
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A few details of the proof
Reverse continuity conditions
Reverse continuity conditions
f
Bδ (i)
A
G(f , µ) and G 0 (A, µ) are “equivalent” in the sense that for each µ,
there exists µ0 (depending only on µ) such that G(f , µ) ⇒ G 0 (A, µ0 )
and vice versa.
D
µ(δ)
Bδ (−i)
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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A few details of the proof
A few details of the proof
Reverse continuity conditions
Strict mutual absolute continuity
G(f , µ) is useful because many of our maps are normalized so that
they fix −i, i, and 1. In order to achieve such a normalization, we
sometimes need to apply a Möbius transformation. The condition
G(f , µ) ensures that such a transformation does not distort distances
too much.
The condition G 0 (A, µ) (typically with A = η or some part of η) is
useful because many of our estimates degenerate near the boundary.
G(f , µ) is well-behaved under compositions of maps. G 0 (A, µ) is easy
to deal with geometrically.
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Almost sure multifractal spectrum of SLE
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The parts of η inside the “pockets” used in the proof of the lower
bound are SLEκ (ρL ; ρR )’s, not ordinary SLEκ ’s.
However, if we grow out a little bit of the curves, then map back, we
get curves whose laws are strictly mutually absolutely continuous with
respect to the law of ordinary SLEκ curve, meaning that their laws are
absolutely continuous, with Radon-Nikodym derivative bounded
above and below by deterministic constants.
We can do this at the same time we move the endpoints to −i and i.
Ewain Gwynne (MIT)
A few details of the proof
Almost sure multifractal spectrum of SLE
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A few details of the proof
Strict mutual absolute continuity
Strict mutual absolute continuity
i
i
η̄
Growing the initial (purple) segments of the curve means that the
starting point for the auxiliary flow lines is not a stopping time for η.
≈ SLEκ
SLEκ (ρL ; ρR )
One has to be careful to make sure that the results of the imaginary
geometry papers are applicable (this actually involves growing a
second pair of auxiliary flow lines).
η
−i
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Almost sure multifractal spectrum of SLE
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Almost sure multifractal spectrum of SLE
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A few details of the proof
A few details of the proof
“o(1)” errors
“o(1)” errors
In our events, we require that the derivative of the conformal map is
“≈ ”, i.e. between s−u and s+u , where u is a small parameter.
In order to ensure that the perfect points are actually contained in the
set Θs (Dη ), we need to shrink u a little bit at each stage.
To counteract the increasing constants in the estimates, we also need
to increase β a little bit at each stage.
The diameter of the nth pocket is e −β n (1+on (1)) , where β n =
Pn
j=1 βj .
To make sure the pockets surrounding z and w are disjoint, we need
to skip non (1) scales when we do the two-point estimate.
This is okay, as it just leads to an on (1) error in the exponent.
So, the nth event in the definition of the perfect points is defined
using the ball of radius e −βn , rather than e −β , where βn → ∞ (at
approximately a logarithmic rate) as n → ∞.
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Almost sure multifractal spectrum of SLE
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Ewain Gwynne (MIT)
Conclusion
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Conclusion
Outline
Future directions
1
Definitions and background
2
Upper bound
3
Lower bound
4
A few details of the proof
5
Conclusion
Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
Winding spectrum of SLE—asymptotics of arg φ0 rather than |φ0 |
Can also consider mixed spectrum.
Predictions by Duplantier and Duplantier/Binder
Upper bound for winding proven by Aru (2014).
Probably lower bound can be done in a similar manner as for the
multifractal spectrum.
Multifractal spectrum for SLEκ (ρ) near where it intersects the
boundary.
Almost sure multifractal spectrum of SLE
Same as for ordinary SLE away from the boundary by absolute
continuity.
Maybe could be done using techniques similar to those of our paper
and/or or those of Alberts-Binder-Viklund.
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Ewain Gwynne (MIT)
Almost sure multifractal spectrum of SLE
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Conclusion
References
E. Gwynne, J. Miller, and X. Sun. Almost sure multifractal spectrum
of SLE. ArXiv.
B Duplantier. Conformally invariant fractals and potential theory.
Physical Review Letters.
G. Lawler and F. J. Viklund. Almost sure multifractal spectrum for
the tip of an SLE curve. Acta Math.
D. Beliaev and S. Smirnov. Harmonic measure and SLE. Comm.
Math. Phys.
S. Sheffield. Conformal weldings of random surfaces: SLE and the
quantum gravity zipper. ArXiv.
J. Miller and S. Sheffield. Imaginary Geometry I-IV. ArXiv.
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Almost sure multifractal spectrum of SLE
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