Large deviations of Schramm-Loewner evolutions: A survey

Yilin Wang

Massachusetts Institute of Technologyyilwang@mit.edu

January 22, 2021

Abstract

These notes survey the first and recent results on large deviations of Schramm-Loewner evolutions (SLE) with the emphasis on interrelations among rate functionsand applications to complex analysis. More precisely, we describe the large deviationsof SLEκ when the κ parameter goes to zero in the chordal and multichorcal caseand to infinity in the radial case. The rate functions, namely Loewner energy andLoewner-Kufarev energy, are of interest from the geometric function theory viewpointand closely related to the Weil-Petersson class of quasicircles and real rational functions.

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Contents

1 Introduction 31.1 Large deviation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Chordal Loewner chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Chordal SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Large deviations of chordal SLE0+ 82.1 Chordal Loewner energy and large deviations . . . . . . . . . . . . . . . . . 82.2 Reversibility of Loewner energy . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Loop energy and Weil-Petersson quasicircles . . . . . . . . . . . . . . . . . . 10

3 Cutting, welding, and flow-lines 133.1 Cutting-welding identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Flow-line identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Large deviations of multichordal SLE0+ 194.1 Multichordal SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Real rational functions and Shapiro’s conjecture . . . . . . . . . . . . . . . 214.3 Large deviations of multichordal SLE0+ . . . . . . . . . . . . . . . . . . . . 224.4 Minimal potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Large deviations of radial SLE∞ 265.1 Radial SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Loewner-Kufarev equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Loewner-Kufarev energy and large deviations . . . . . . . . . . . . . . . . . 28

6 Foliation of Weil-Petersson quasicircles 296.1 Whole-plane Loewner evolution . . . . . . . . . . . . . . . . . . . . . . . . . 306.2 Energy duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7 Summary 33

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1 Introduction

The Schramm-Loewner evolution (SLE) is a model for conformally invariant random fractalcurves in the plane, introduced by Schramm [Sch00] by combining Loewner’s classicaltheory [Loe23] for the evolution of planar slit domains with stochastic analysis. Schramm’sSLE is a one-parameter family of probability measures on non-self-crossing curves, indexedby κ ≥ 0, and thus denoted by SLEκ. When κ > 0, these curves are fractal, andthe parameter κ reflects the curve’s roughness. SLEs play a central role in the recentdevelopment of 2D random conformal geometry. For instance, they describe interfaces inconformally invariant systems arising from scaling limits of discrete models in statisticalphysics, which was also Schramm’s original motivation, see, e.g., [LSW04,Sch07,Smi06,SS09].Through their relationship with critical statistical physics models, SLEs are also closelyrelated to conformal field theory (CFT) whose central charge is a function of κ, see,e.g., [BB03,Car03,FW03,FK04,Dub15,Pel19].Large deviation principle describes the probability of rare events of a given family ofprobability measures on an exponential scale. The formalization of the general frameworkof large deviation was developed by (or commonly attributed to) Varadhan [Var66]. Agreat deal of mathematics has been developed since then. Large deviations estimates haveproved to be the crucial tool required to handle many questions in statistics, engineering,statistical mechanics, and applied probability.In these notes, we give a minimalist account of basic definitions and ideas from boththe SLE and large deviation theory, only sufficient for considering the large deviations ofSLE. We by no means attempt to give a thorough reference to the background of thesetwo theories. Our approach focuses on how large deviation consideration propels to thediscovery (and rediscovery) of interesting deterministic objects, including Loewner energy,Loewner-Kufarev energy, Weil-Petersson quasicircles, real rational functions, foliations, etc.,and the interplay among them. Unlike the fractal or discrete nature of objects consideredin random conformal geometry, these deterministic objects, arising from the κ → 0+ or∞ large deviations (on which the rate function is finite), are more regular. However, theystill capture the essence of many coupling results in random conformal geometry. As forthe large deviation literature, these results also show new ways to apply this powerfulmachinery. We summarize and compare the quantities and theorems from both randomconformal geometry and the finite energy world in the last section. Impatient readers mayskip to the end to get a flavor about the closeness of the analogy.The main theorems presented here are collected from [Wan19a,Wan19b,RW19,VW20a,PW20,APW20,VW20b]. Compared to those research papers, we choose to outline theintuition behind the theorems and sometimes omit proofs or only present the proof in asimpler case to illustrate the idea. These lecture notes are written based on two lectureseries that I gave at the joint webinar of Tsinghua-Peking-Beijing Normal Universities andRandom Geometry and Statistical Physics online seminars during the Covid-19 time. Iwould like to thank the organizers for the invitation and the online lecturing experienceunder the pandemic’s unusual situation and is supported by the NSF grant DMS-1953945.

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1.1 Large deviation principle

We consider first a simple example to illustrate the concept of large deviations. LetX ∼ N (0, σ2) be a real, centered Gaussian random variable of variance σ2. The densityfunction of X is given by

pX(x) =1√

2πσ2exp(− x

2

2σ2 ).

Let ε > 0,√εX ∼ N (0, σ2ε). As ε → 0+,

√εX converges almost surely to 0, so the

probability measure p√εX on R converges to the Dirac measure δ0.Let M > 0. We are interested in the rare event {

√εX ≥M} which has probability

P(√εX ≥M) = 1√

2πσ2ε

∫ ∞M

exp(− x2

2σ2ε)dx.

To quantify how rare this event happens when ε→ 0+, we have

ε logP(√εX ≥M) = ε log

(1√

2πσ2ε

∫ ∞M

exp(− x2

2σ2ε)dx)

= −12ε log(2πσ2ε) + ε log

∫ ∞M

exp(− x2

2σ2ε)dx

ε→0+−−−−→ −M2

2σ2 =: −IX(M) = − infx∈[M,∞)IX(x).

The function IX : x 7→ x2/2σ2 is called the large deviation rate function of the family ofrandom variables (

√εX)ε>0.

Now let us state the large deviation principle more precisely. Let X be a Polish space, Bits Borel σ-algebra, {µε}ε>0 a family of probability measures on (X ,B). To reduce possiblemeasurability questions, all probability spaces are assumed to have been completed.

Definition 1.1. A rate function is a lower semicontinuous mapping I : X → [0,∞], namely,for all α ≥ 0, the sub-level set {x : I(x) ≤ α} is a closed subset of X . A good rate functionis a rate function for which all the sub-level sets are compact subsets of X .

Definition 1.2. We say that a family of probability measures {µε}ε>0 on (X ,B) satisfiesthe large deviation principle of rate function I if for all open sets O ∈ B and closed setF ∈ B,

limε→0+

ε logµε(O) ≥ − infx∈O

I(x); limε→0+

ε logµε(F ) ≤ − infx∈F

I(x).

Note that if a large deviation rate function exists then it is unique, see, e.g., [Din93, Lem. 1.1].

Remark 1.3. If A ∈ B and infx∈Ao I(x) = infx∈A I(x) (we call such Borel set a continuityset of I), then the large deviation principle gives

limε→0+

ε logµε(A) = − infx∈A

I(x).

Remark 1.4. It is not hard to check that the distribution of {√εX}ε>0 satisfies the large

deviation principle with good rate function IX .

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The reader should mind carefully that large deviation results depend on the topologyinvolved which can be a subtle point. On the other hand, it follows from the definitionthat the large deviation principle transfers nicely through continuous functions:

Theorem 1.5 (Contraction principle [DZ10, Thm. 4.2.1]). If X ,Y are two Polish spaces,θ : X → Y a continuous function {µε}ε>0 a family of probability measures on X , satisfyingthe large deviation principle with good rate function I : X → [0,∞]. Let I ′ : Y → [0,∞] bedefined as

I ′(y) := infx∈θ−1{y}

I(x).

Then the family of pushforward probability measures (θ∗µε)ε>0 on Y satisfies the largedeviation principle with good rate function I ′.

The next result is the large deviation principle of the scaled Brownian path. Let T ∈ (0,∞].We recall that the Dirichlet energy of a real-valued function (t 7→Wt) ∈ C0[0, T ] is

IT (W ) :=12

∫ T0|∂tWt|2 dt, (1.1)

if W is absolutely continuous, and set to equal ∞ otherwise. Equivalently, we can write

IT (W ) = supΠ

k−1∑i=0

(Wti+1 −Wti)2

2(ti+1 − ti)(1.2)

where the supremum is taken over all k ∈ N and all partitions {0 = t0 < t1 < · · · < tk ≤ T}.Note that the sum on the right-hand side of (1.2) is the Dirichlet energy of the linearinterpolation of W from its values at (t0, · · · , tk) which is set to be constant on [tk, T ].

Theorem 1.6. (Schilder; see, e.g., [DZ10, Ch. 5.2]) Fix T ∈ (0,∞). The family of processes{(√εBt)t∈[0,T ]}ε>0 satisfies the following large deviation principle in (C0[0, T ], ‖·‖∞) with

good rate function IT .

Remark 1.7. We note that Brownian path has almost surely infinite Dirichlet energy, i.e.,IT (B) =∞. In fact, W has finite Dirichlet energy implies that W is 1/2-Hölder, whereasBrownian motion is only a.s. (1/2− δ)-Hölder for δ > 0. However, Schilder’s theorem showsthat Brownian motion singles out the Dirichlet energy which quantifies, as ε → 0+, thedensity of Brownian path around a deterministic function W . In fact, let Oδ(W ) be the δneighborhood of W in C0[0, T ]. From the monotonicity of infW̃∈Oδ(W ) IT (W̃ ) in δ, Oδ(W )is a continuity set for IT with exceptions for at most countably many δ. Hence, we have

−ε logP(√εB ∈ Oδ(W ))

ε→0+−−−−→ infW̃∈Oδ(W )

IT (W̃ )δ→0+−−−−→ IT (W ). (1.3)

Or, written with some abuse:

P(√εB stays close to W ) ∼ε→0+ exp(−IT (W )/ε). (1.4)

We now give some heuristics to show that the Dirichlet energy appears naturally as thelarge deviation rate function of the scaled Brownian motion. Fix 0 = t0 < t1 < . . . < tk ≤ T .The finite dimensional marginals of the Brownian motion (Bt0 , . . . , Btk) gives a family

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of independent Gaussian random variables (Bti+1 −Bti)0≤i≤k−1 with variances (ti+1 − ti)respectively. Multiplying the Gaussian vector by

√ε, we obtain the large deviation principle

of the finite dimensional marginal with rate function∑k−1i=0

(Wti+1−Wti )2

2(ti+1−ti) from Remark 1.4,Theorem 1.5, and the independence of the family of increments. Approximating theBrownian motion on the finite interval [0, T ] by the linearly interpolated function from itsvalue at times ti, it suggests that the scaled Brownian paths satisfy the large deviationprinciple of rate function the supremum of the rate function of all of its finite dimensionalmarginals which then turns out to be the Dirichlet energy by (1.2).A rigorous proof of Schilder’s theorem uses the Cameron-Martin theorem which allowsgeneralization to any abstract Wiener space. Namely, the associated family of Gaussianmeasures scaled by

√ε satisfies the large deviation principle with rate function 1/2 times

its Cameron-Martin norm. See, e.g., [DS89, Thm. 3.4.12].

Remark 1.8. Dawson-Gärtner theorem (see, e.g., [DZ10, Thm. 4.6.1]) on the large devia-tion principle of projective limits shows that Schilder’s theorem also holds for the infinitetime Brownian motion, when C0[0,∞) is endowed with the topology of uniform convergenceon compact sets (since it is the projective limit of C0[0, T ] as T →∞).

1.2 Chordal Loewner chain

The description of SLE is based on the Loewner transform, a deterministic procedure thatencodes a non-self-intersecting curve on a 2-D domain into a driving function. In thissurvey, we use two types of Loewner chain: the chordal Loewner chain in (D;x, y), whereD is a simply connected domain with two distinct boundary points x (starting point) andy (target point); and later in Section 5, the radial Loewner chain in D targeting at aninterior point. The definition is invariant under conformal maps (biholomorphic functions).Hence by Riemann mapping theorem it suffices to describe in the chordal case when(D;x, y) = (H; 0,∞), and in the radial case when D = D, targeting at 0. Throughout thearticle, H = {z ∈ C : Im(z) > 0} is the upper halfplane, H∗ = {z ∈ C : Im(z) < 0} is thelower halfplane, D = {z ∈ C : |z| < 1} is the unit disk, and D∗ = {z ∈ C : |z| > 1} ∪ {∞}.Let us start with this chordal Loewner description of a continuous simple curve γ from0 to ∞ in H. We parameterize the curve by the halfplane capacity. More precisely, γis continuously parametrized by R+, with γ0 = 0, γt → ∞ as t → ∞, in the way suchthat the unique conformal map gt from H r γ[0,t] onto H with the expansion at infinitygt(z) = z + o(1) satisfies

gt(z) = z + 2t/z + o(1/z). (1.5)

The coefficient 2t is the halfplane capacity of γ[0,t]. One shows that gt can be extendedby continuity to the boundary point γt and that the real-valued function Wt := gt(γt) iscontinuous. This function W is called the driving function of γ.Conversely, the chordal Loewner chain in (H; 0,∞) driven by a continuous real-valuedfunction W ∈ C0[0,∞) is the family of conformal maps (gt)t≥0, obtained by solving theLoewner equation for each z ∈ H,

∂tgt(z) =2

gt(z)−Wtwith initial condition g0(z) = z. (1.6)

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The solution t 7→ gt(z) to (1.6) is defined up to the swallowing time

τ(z) := sup{t ≥ 0: infs∈[0,t]

|gs(z)−Ws| > 0}

of z, and we obtain a family of growing hulls Kt = {z ∈ H : τ(z) ≤ t}. The solution gt of(1.6) satisfies the expansion (1.5) near ∞ and is the unique conformal map from HrKtonto H with expansion z+ o(1) as z →∞. Moreover, Kt has halfplane capacity 2t. ClearlyKt and gt uniquely determine each other. We list a few properties of the Loewner chain.

• If W is the driving function of a simple chord γ, then we have Kt = γ[0,t], and thesolution gt of (1.6) is exactly the conformal map constructed from γ as in (1.5).• The imaginary axis iR+ is driven by W ≡ 0.• (Additivity) Let (Kt)t≥0 be the family of hulls generated by the driving function W .

Fix s > 0, the driving function generating (gs(Kt+srKs)−Ws)t≥0 is t 7→Ws+t−Ws.• (Scaling) Fix λ > 0, the driving function generating the scaled (and reparameterized)family of hulls (λKλ−2t)t≥0 is t 7→ λWλ−2t.• Not every continuous driving function arises from a simple chord. It is unknownhow to characterize analytically the class of functions which generate simple curves.Sufficient conditions exist, such as when W is 1/2-Hölder with Hölder norm strictlyless than 4 [MR05,Lin05].

1.3 Chordal SLE

We now very briefly review the definition and relevant properties of chordal SLE. Forfurther SLE background, we refer the readers to, e.g., [Law05,Wer04b]. The chordalSchramm-Loewner evolution of parameter κ in (H; 0,∞), denoted by SLEκ, is the randomnon-self-intersecting curve tracing out the hulls (Kt)t≥0 generated by

√κB via the Loewner

transform, where B is the standard Brownian motion and κ ≥ 0. SLEs describe theinterfaces in the scaling limit of various statistical mechanics models, e.g.,

• SLE2 ↔ Loop-erased random walk [LSW04];• SLE8/3 ↔ Self-avoiding walk (conjecture);• SLE3 ↔ Critical Ising model interface [Smi10];• SLE4 ↔ Level line of the Gaussian free field [SS09];• SLE6 ↔ Critical independent percolation interface [Smi06];• SLE8 ↔ Contour line of uniform spanning tree [LSW04].

The reason that SLE curves describe those interfaces arising from conformally invariantsystems is that they are the unique random Loewner chain that are scaling-invariant andsatisfy the domain Markov property. More precisely, for λ > 0, the law is invariant underthe scaling transformation (then reparametrize by capacity)

(Kt)t≥0 7→ (Kλt := λKλ−2t)t≥0

and for all s ∈ [0,∞), if one defines K(s)t = gs(Ks+trKs)−Ws, where (Kt) is driven by W ,then (K(s)t )t≥0 has the same distribution as (Kt)t≥0 and is independent of σ(Wr : r ≤ s).

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In fact, these two properties on (Kt) translate into the independent stationary incrementsproperty (i.e., being a continuous Lévy process) and the invariance under Wt λWλ−2tof the driving function. Multiples of Brownian motions are the only continuous processessatisfying these two properties.The scaling-invariance makes it possible to define SLE in other simply connected domains(D;x, y) as the preimage of SLE in (H; 0,∞) by a conformal map ϕ : D → H takingrespectively the boundary points x, y to 0,∞ (another choice of ϕ̃ equals λϕ for someλ > 0). The chordal SLE is therefore conformally invariant from the definition.

Remark 1.9. The SLE0 in (H; 0,∞) is simply the Loewner chain driven by W ≡ 0,namely the imaginary axis iR+. It implies that the SLE0 in (D;x, y) equals ϕ(iR+) (i.e.,the hyperbolic geodesic in D connecting x and y).

SLE curves exhibit phase transitions depending on the value of κ:

Theorem 1.10 ([RS05]). For κ ∈ [0, 4], SLEκ is almost surely a simple curve starting at0. For κ ∈ (4, 8), SLEκ is almost surely traced out by a self-touching curve. For κ ∈ [8,∞),SLEκ is almost surely a space-filling curve. Moreover, for all κ ≥ 0, the trace of SLE goesto ∞ as t→∞ almost surely.

2 Large deviations of chordal SLE0+

2.1 Chordal Loewner energy and large deviations

We first specify the topology on simple chords in our large deviation result (Theorem 2.4).

Definition 2.1. The Hausdorff distance dh of two compact sets F1, F2 ⊂ D is defined as

dh(F1, F2) := inf{ε ≥ 0

∣∣∣ F1 ⊂ ⋃x∈F2

Bε(x) and F2 ⊂⋃x∈F1

Bε(x)},

where Bε(x) denotes the Euclidean ball of radius ε centered at x ∈ D. We then define theHausdorff metric on the set of closed subsets of D via the pullback by a conformal mapD → D. Although the metric depends on the choice of the conformal map, the topologyinduced by dh is canonical, as conformal automorphisms of D are fractional linear functions,that are uniformly continuous on D. We endow the space X (D;x; y) of unparametrizedsimple chords in (D;x; y) with the relative topology induced by the Hausdorff metric.

Definition 2.2. The Loewner energy of a chord γ ∈ X (D;x, y) is defined as the Dirichletenergy (1.1) of its driving function,

ID;x,y(γ) := IH;0,∞(ϕ(γ)) := I∞(W ), (2.1)

where ϕ is any conformal map from D to H such that ϕ(x) = 0 and ϕ(y) =∞, W is thedriving function of ϕ(γ) and I∞(W ) is the Dirichlet energy as defined in (1.1) and (1.2).

Note that the definition of ID;x,y(γ) does not depend on the choice of ϕ. In fact, since twochoices differ only by post-composing by a scaling factor. From the scaling property of the

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Loewner driving function,W changes to t 7→ λWλ−2t which has the same Dirichlet energy asW . The Loewner energy ID;x,y(γ) is non-negative and minimized by the hyperbolic geodesicsince the driving function of ϕ(η) is just the constant function W ≡ 0, so ID;x,y(η) = 0.

Theorem 2.3 ([Wan19a, Prop. 2.1]). If I∞(W ) 0, with ε = κ. Indeed, the following result is proved in [PW20] which strengthensa similar result in [Wan19a]. As we are interested in the 0+ limit, we only consider small κwhere the trace γκ := ∪t≥0Kt of SLEκ is almost surely in X (D;x, y).

Theorem 2.4 ([PW20, Thm. 1.4]). The family of distributions {Pκ}κ>0 on X (D;x, y) ofthe chordal SLEκ curves satisfies the large deviation principle with good rate function ID;x,y.That is, for any open set O and closed set F of X (D;x, y), we have

limκ→0+

κ logPκ[γκ ∈ O] ≥ − infγ∈O

ID;x,y(γ),

limκ→0+

κ logPκ[γκ ∈ F ] ≤ − infγ∈F

ID;x,y(γ),

and the sub-level set {γ ∈ X (D;x, y) | ID;x,y(γ) ≤ c} is compact for any c ≥ 0.

We note that the Loewner transform mapping continuous driving function to the union ofhulls it generates, endowed with the Hausdorff metric, is not continuous and the contractionprinciple (Theorem 1.5) does not apply. Therefore Schilder’s theorem does not implytrivially the large deviation principle of SLE0+. This result thus requires some work,see [PW20] for more details.

Remark 2.5. As Remark 1.7, we emphasize that finite energy chords are more regularthan SLEκ curves for any κ > 0. In fact, as we will see in Theorem 2.14, finite energy chordis part of a Weil-Petersson quasicircle which is rectifiable, see Remark 2.17. On the otherhand, Beffara [Bef08] shows that for 0 < κ ≤ 8, SLEκ has Hausdorff dimension 1 + κ/8 > 1,thus even the length of a piece of SLEκ curve is not defined when κ > 0.

2.2 Reversibility of Loewner energy

Given that SLEκ curves arise as scaling limits of interfaces in lattice models from statisticalphysics, it was natural to conjecture that chordal SLE is reversible, i.e., the time-reversal ofa chordal SLEκ from a to b in D is a chordal SLEκ from b to a in D after reparametrization,at least for specific κ values. It was first settled by Zhan [Zha08b] in the case of the simplecurves κ ∈ [0, 4] via rather non-trivial couplings of both ends of the SLE path. See alsoDubédat’s commutation relations [Dub07], and Miller and Sheffield’s approach based on theGaussian Free Field [MS16a,MS16b,MS16c] that also provides a proof in the non-simplecurve case when κ ∈ (4, 8].

Theorem 2.6 (SLE reversibility [Zha08b]). For κ ∈ [0, 4], the distribution of the trace γκof SLEκ in (H, 0,∞) coincides with that of its image under ι : z → −1/z.

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We deduce from Theorem 2.4 and Theorem 2.6 the following result.

Theorem 2.7 (Energy reversibility [Wan19a]). We have ID;x,y(γ) = ID,y,x(γ) for anychord γ ∈ X (D;x, y).

Proof. Without loss of generality, we assume that (D;x, y) = (H; 0,∞). We want to showthat IH;0,∞(γ) = IH;0,∞(ι(γ)).Let δn be a sequence of numbers converging to 0+, such that Oδn(γ) = {γ̃ ∈ X (H; 0,∞) :θ∗dh(γ, γ̃) < δn} is a continuity set for IH;0,∞. The sequence exists since there are at mosta countable number of exceptions of δ such that Oδ(γ) is not a continuity set. In fact, forδ < δ′, we have the inclusion Oδ(γ) ⊂ Oδ(γ) ⊂ Oδ′(γ) and

infγ̃∈Oδ′ (γ)

IH;0,∞(γ̃) ≤ infγ̃∈Oδ(γ)

IH;0,∞(γ̃) ≤ infγ̃∈Oδ(γ)

IH;0,∞(γ̃).

Therefore the function inf γ̃∈Oδ(γ) IH;0,∞(γ̃) has a jump at points δ > 0 for which Oδ(γ) isnot a continuity set. The monotonicity of δ 7→ inf γ̃∈Oδ(γ) IH;0,∞(γ̃) shows that we have atmost countably many such discontinuity points. From Remark 1.3 and Theorem 2.6, wehave

limκ→0+

κ logPκ(γκ ∈ Oδn(γ)) = − infγ̃∈Oδn (γ)

IH;0,∞(γ̃), (2.2)

which tends to −IH;0,∞(γ) as n→∞ from the lower-semicontinuity of IH;0,∞.We now specify the conformal map ϑ : H→ D sending i to 0 to define the Hausdorff metricin H as in Definition 2.1. This choice makes ϑ ◦ ι ◦ ϑ−1 = − IdD. In other words, ι inducesan isometry on closed sets of H. Theorem 2.6 then shows that

Pκ(γκ ∈ Oδn(γ)) = Pκ(ι(γκ) ∈ ι(Oδn(γ))) = Pκ(γκ ∈ Oδn(ι(γ))).

We obtain the claimed energy reversibility by applying (2.2) to ι(γ).

Remark 2.8. This proof presented above is different from [Wan19a] but very close in thespirit. We used here Theorem 2.4 from the recent work [PW20], whereas the original proofin [Wan19a] used the more complicated left-right passing events without the strong largedeviation result at hand.

Remark 2.9. The energy reversibility is a result about deterministic chords. However, fromthe definition alone, the reversibility is not obvious as the Loewner evolution is directional.To illustrate it, consider the example of a driving function W with finite Dirichlet energythat is constant after time 1 (and contributes 0 energy). From the additivity property ofdriving function, γ[1,∞) is the hyperbolic geodesic in Hr γ[0,1]. The reversed curve ι(γ) is achord starting with an analytic curve which deviates from the imaginary axis. Thereforeunlike γ, the energy of ι(γ) typically spreads over the whole time interval R+.

2.3 Loop energy and Weil-Petersson quasicircles

We now generalize the Loewner energy to Jordan curves (simple loops) on the Riemannsphere Ĉ = C ∪ {∞}. This generalization will show that the Loewner energy exhibits more

10

0γ

z 7→ z2

0∞0

γ

H

C \ R+

Figure 1: From chord in (H; 0,∞) to a Jordan curve.

symmetries (Theorem 2.10). Moreover, an equivalent description (Theorem 2.14) of theloop energy will give an analytic proof of those symmetries including the reversibility.Let γ : [0, 1]→ Ĉ be a parametrized Jordan curve with the marked point γ(0) = γ(1). Forevery ε > 0, γ[ε, 1] is a chord connecting γ(ε) to γ(1) in the simply connected domainĈ r γ[0, ε]. The rooted loop Loewner energy of γ rooted at γ(0) is defined as

IL(γ, γ(0)) := limε→0

IĈrγ[0,ε],γ(ε),γ(0)(γ[ε, 1]).

The loop energy generalizes the chordal energy. In fact, let η be a simple chord in(Cr R+, 0,∞) and we parametrize γ = η ∪ R+ in a way such that γ[0, 1/2] = R+ ∪ {∞}and γ[1/2, 1] = η. Then from the additivity of chordal energy,

IL(γ,∞) =ICrR+,0,∞(η) + limε→0 IĈrγ[0,ε],γ(ε),γ(0)(γ[ε, 1/2])

=ICrR+,0,∞(η),

since γ[ε, 1/2] is contained in the hyperbolic geodesic between γ(ε) and γ(0) in Ĉ r γ[0, ε]for all 0 < ε < 1/2, see Figure 1. Steffen Rohde and I proved the following result.

Theorem 2.10 ([RW19]). The loop energy does not depend on the root.

We do not present the original proof of this theorem since it will follow immediate fromTheorem 2.14, see Remark 2.16.

Remark 2.11. From the definition, the loop energy IL is invariant under the Möbiustransformations of Ĉ, and IL(γ) = 0 if and only if γ is a circle (or a line).

Remark 2.12. The loop energy is presumably the large deviation rate function of SLE0+loop measure on Ĉ constructed in [Zha20] (see also [Wer08,BD16] for the constructionwhen κ = 8/3 and 2). However, the conformal invariance of the SLE loop measures impliesthat they have infinite total mass and has to be normalized properly for considering largedeviations. We do not claim it here and think it is an interesting question to work out.However, these ideas will serve as heuristics to deduce results for finite energy loops inSection 3.

In [RW19] we also showed that if a Jordan curve has finite energy, then it is a quasicircle,namely the image of a circle or a line under a quasiconformal map of C (and a quasiconformalmap is a homeomorphism that maps locally small circles to ellipses with uniformly bounded

11

eccentricity). However, not all quasicircles have finite energy since they may have Hausdorffdimension larger than 1. The natural question is then to identify the family of finite energyquasicircles. The answer is surprisingly a family of so-called Weil-Petersson quasicircles,which has been studied by both physicists and mathematicians since the eighties, includingBowick, Rajeev, Witten, Nag, Verjovsky, Sullivan, Cui, Tahktajan, Teo, Shen, Sharon,Mumford, Pommerenke, González, Bishop, etc., and is still an active research area. See theintroduction of the recent preprint [Bis19] for a summary and a list of (currently) morethan twenty equivalent definitions of very different nature.We will use the following definition. The class of Weil-Petersson quasicircles is preservedunder Möbius transformation, so without loss of generality, we only spell out the definitionof a bounded Weil-Petersson quasicircle. Let γ be a bounded Jordan curve, and write Ωand Ω∗ be the connected components of Ĉ r γ. Let f be a conformal map from D onto thebounded component Ω, and h a conformal map from D∗ onto the unbounded componentΩ∗ fixing ∞.

Definition 2.13. The bounded Jordan curve γ is a Weil-Petersson quasicircle if and onlyif the following equivalent conditions hold:

1. DD(log |f ′|) := 1π∫D |∇ log |f ′| (z)|

2 dA(z) = 1π∫D |f ′′(z)/f ′(z)|

2 dA(z)

then shows that Weil-Petersson quasicircles are asymptotically smooth, namely, chord-arcwith local constant 1: for all x, y on the curve, the shorter arc γx,y between x and y satisfies

lim|x−y|→0

length (γx,y)/|x− y| = 1.

(Chord-arc means length(γx,y)/|x− y| is uniformly bounded.)

The connection between Loewner energy and Weil-Petersson quasicircles goes further: Notonly the Loewner energy identifies Weil-Petersson quasicircles from its finiteness, but isalso closely related to the Kähler structure on T0(1), the Weil-Petersson Teichmüller space,naturally associated to the class of Weil-Petersson quasicircles. In fact, the right-handside of (2.3) coincides with the universal Liouville action introduced by Takhtajan andTeo [TT06] and shown by them to be a Kähler potential of the Weil-Petersson metric, whichis the unique homogeneous Kähler metric on T0(1) up to a scaling factor. To summarize,we obtain:

Corollary 2.18. The Loewner energy is a Kähler potential of the Weil-Petersson metricon T0(1).

We do not enter into further details as it goes beyond the scope of large deviations that wechoose to focus on here. Another reason is that this unexpected link still lacks a betterexplanation and we definitely need to understand further the relation between SLEs andthe Kähler structure on T0(1).

3 Cutting, welding, and flow-lines

Pioneering works [Dub09b,She16,MS16a] on couplings between SLEs and Gaussian freefield (GFF) have led to many remarkable applications in the study of 2D random conformalgeometry. These couplings are often speculated from the link to the discrete models.In [VW20a], Viklund and I provided another view on these couplings through the lens oflarge deviations by showing the interplay between Loewner energy of curves and Dirichletenergy of functions defined in the complex plane (which is the large deviation rate functionof scaled GFF). These results are analogous to the SLE/GFF couplings, but the proofs areremarkably short and use only analytic tools without any of the probabilistic models.

3.1 Cutting-welding identity

To state the result, we write E(Ω) for the space of real functions on a domain Ω ⊂ C withweak first derivatives in L2(Ω) and recall the Dirichlet energy of ϕ ∈ E(Ω) is

DΩ(ϕ) :=1π

∫Ω|∇ϕ|2dA(z).

Theorem 3.1 (Cutting [VW20a, Thm. 1.1]). Suppose γ is a Jordan curve through ∞ andϕ ∈ E(C). Then we have the identity:

DC(ϕ) + IL(γ) = DH(u) +DH∗(v), (3.1)

13

whereu = ϕ ◦ f + log

∣∣f ′∣∣ , v = ϕ ◦ h+ log ∣∣h′∣∣ , (3.2)where f and h map conformally H and H∗ onto, respectively, H and H∗, the two componentsof Cr γ, while fixing ∞.

The function ϕ ∈ E(C) implies that it has vanishing mean oscillation. The John-Nirenberginequality (see, e.g., [Gar07, Thm.VI.6.4]) shows that e2ϕ is locally integrable and strictlypositive. In other words, e2ϕdA(z) defines a σ-finite measure supported on C, absolutelycontinuous with respect to Lebesgue measure dA(z). The transformation law (3.2) ischosen such that e2udA(z) and e2vdA(z) are the pullback measures by f and h of e2ϕdA(z),respectively.We explain first why we consider this theorem as a deterministic analog of an SLE/GFFcoupling. Note that we do not make rigorous statement here and only argue heuristically.In fact, Sheffield’s quantum zipper theorem couples SLEκ curves with quantum surfacesvia a cutting operation and as welding curves [She16,DMS14]. A quantum surface is adomain equipped with a Liouville quantum gravity (

√κ-LQG) measure, defined using

a regularization of e√κΦdA(z), where

√κ ∈ (0, 2), and Φ is a Gaussian field with the

covariance of a free boundary GFF. The analogy is outlined in the table below.

SLE/GFF with κ� 1 Finite energySLEκ loop Jordan curve γ with IL(γ)

On the other hand the independence between Φ1 and Φ2 gives

“ limκ→0−κ logP(

√κΦ1 stays close to 2u,

√κΦ2 stays close to 2v)

= DH(u) +DH∗(v)”.

We obtain the identity (3.1) heuristically.We now present our short proof of Theorem 3.1 in the case where γ is smooth andϕ ∈ C∞c (C) to illustrate the idea. The general case follows from an approximationargument, see [VW20a] for the complete proof.

Proof. From Remark 2.15, if γ passes through ∞, then

IL(γ) = 1π

∫H|∇σf |2 dA(z) +

1π

∫H∗|∇σh|2 dA(z),

where σf and σh are the shorthand notation for log |f ′| and log |h′|. The conformal invarianceof Dirichlet energy gives

DH(ϕ ◦ f) +DH∗(ϕ ◦ h) = DH(ϕ) +DH∗(ϕ) = DC(ϕ).

To show (3.1), after expanding the Dirichlet energy terms, it suffices to verify the crossterms that∫

H〈∇σf (z),∇(ϕ ◦ f)(z)〉 dA(z) +

∫H∗〈∇σh(z),∇(ϕ ◦ h)(z)〉dA(z) = 0. (3.3)

By Stokes’ formula, the first term on the left-hand side equals∫R∂nσf (x)ϕ(f(x))dx =

∫Rk(f(x))

∣∣f ′(x)∣∣ϕ(f(x))dx = ∫∂H

k(z)ϕ(z) |dz|

where k(z) is the geodesic curvature of γ = ∂H at z using the identity ∂nσf (x) =|f ′(x)|k(f(x)) (this follows from an elementary differential geometry computation, see,e.g., [Wan19b, Appx.A]). The geodesic curvature at the same point z ∈ γ, considered asa point of ∂H∗, equals −k(z). Therefore the sum in (3.3) cancels out and completes theproof in the smooth case.

The following result is on the converse operation of the cutting, which shows that we canalso recover γ and ϕ from u and v by conformal welding. More precisely, an increasinghomeomorphism w : R→ R is said to be a (conformal) welding homeomorphism of a Jordancurve γ through ∞, if there are conformal maps f, h of the upper and lower half-planesonto the two components of Cr γ, respectively, such that w = h−1 ◦ f |R. Suppose H andH∗ are each equipped with an infinite and positive boundary measure defining a metricon R such that the distance between x < y equals the measure of [x, y]. Under suitableassumptions, the increasing isometry w : R = ∂H→ ∂H∗ = R fixing 0 is well-defined and awelding homeomorphism of some curve γ. In this case, we say that the corresponding tuple(γ, f, h) solves the isometric welding problem for the given measures.

Theorem 3.2 (Isometric conformal welding [VW20a, Thm. 1.2]). Suppose u ∈ E(H) andv ∈ E(H∗) are given. The isometric welding problem for the measures eudx and evdx on Rhas a solution (γ, f, h) and the welding curve γ has finite Loewner energy. Moreover, thereexists a unique ϕ ∈ E(C) such that (3.2) is satisfied.

15

In the statement, the measures eudx and evdx are defined using the traces of u, v on R,that are H1/2(R) functions. We say that u ∈ H1/2(R) if

‖u‖2H1/2 :=1

2π2∫∫

R×R

|u(x)− u(y)|2

|x− y|2dx dy

Theorem 3.4 (Flow-line identity [VW20a, Thm. 1.4]). Let ϕ ∈ E(C) ∩ C0(Ĉ). Any flow-line γ of the vector field eiϕ is a Jordan curve through ∞ with finite Loewner energy andwe have the formula

DC(ϕ) = IL(γ) +DC(ϕ0). (3.6)

Remark 3.5. The assumption of ϕ ∈ C0(Ĉ) is for technical reason to consider the flow-lineof eiϕ in the classical differential equation sense. One may also drop this assumption bydefining a flow-line to be a chord-arc curve γ passing through ∞ on which ϕ = τ arclengthalmost everywhere. We will further explore these ideas in a setting adapted to boundedcurves (see Theorem 6.13).

This identity is analogous to the flow-line coupling between SLE and GFF, of criticalimportance, e.g., in the imaginary geometry framework of Miller-Sheffield [MS16a]: veryloosely speaking, an SLEκ curve may be coupled with a GFF Φ and thought of as a flow-lineof the vector field eiΦ/χ, where χ = 2/γ − γ/2. As γ → 0, we have eiΦ/χ ∼ eiγΦ/2.Let us finally remark that by combining the cutting-welding (3.1) and flow-line (3.6)identities, we obtain the following complex identity. See also Theorem 6.13 the complexidentity for a bounded Jordan curve.

Corollary 3.6 (Complex identity [VW20a, Cor. 1.6]). Let ψ be a complex-valued functionon C with finite Dirichlet energy and whose imaginary part is continuous in Ĉ. Let γ be aflow-line of the vector field eψ. Then we have

DC(ψ) = DH(ζ) +DH∗(ξ),

where ζ = ψ ◦ f + (log f ′)∗, ξ = ψ ◦ h+ (log h′)∗ and z∗ is the complex conjugate of z.

Remark 3.7. A flow-line γ of the vector field eψ is understood as a flow-line of ei Imψ, asthe real part of ψ only contributes to a reparametrization of γ.

Proof. From the identity arg f ′ = P[Imψ] ◦ f , we have

ζ =(Reψ ◦ f + log |f ′|

)+ i(Imψ ◦ f − arg f ′

)= u+ i Imψ0 ◦ f ;

ξ = v + i Imψ0 ◦ h,

where u := Reψ ◦ f + log |f ′| ∈ E(H), v := Reψ ◦ h+ log |h′| ∈ E(H∗) and ψ0 = ψ−P [ψ|γ ].From the cutting-welding identity (3.1), we have

DC(Reψ) + IL(γ) = DH(u) +DH∗(v).

On the other hand, the flow-line identity gives DC(Imψ) = IL(γ) +DC(Imψ0). Hence,

DC(ψ) = DC(Reψ) +DC(Imψ) = DC(Reψ) + IL(γ) +DC(Imψ0)= DH(u) +DH∗(v) +DC(Imψ0)= DH(ζ) +DH∗(ξ)

as claimed.

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Remark 3.8. From Corollary 3.6 we can easily recover the flow-line identity (Theorem 3.4),by taking Imψ = ϕ and Reψ = 0. Similarly, the cutting-welding identity (3.1) followsfrom taking Reψ = ϕ and Imψ = P[τ ] where τ is the winding function along the curve γ.Therefore, the complex identity is equivalent to the union of cutting-welding and flow-lineidentities.

3.3 Applications

We now show that these identities between Loewner and Dirichlet energies inspired byprobabilistic couplings, have interesting consequences in geometric function theory.Suppose γ1, γ2 are locally rectifiable Jordan curves of the same length (possibly bothinfinite) bounding two domains Ω1 and Ω2 and we mark a point on each curve. Let wbe an arclength isometry γ1 → γ2 matching the marked points. We obtain a topologicalsphere from Ω1 ∪ Ω2 by identifying the matched points. Following Bishop [Bis90], theisometric welding problem is to find a Jordan curve γ ⊂ Ĉ, and conformal equivalencesf1, f2 from Ω1 and Ω2 to the two connected components of Ĉr γ, such that f−12 ◦ f1|γ1 = w.The welding problem is in general a hard question and have many pathological examples.For instance, the mere rectifiability of γ1 and γ2 does not guarantee the existence northe uniqueness of γ, but the chord-arc property does. However, chord-arc curves are notclosed under isometric conformal welding: the welding curve can have Hausdorff dimensionarbitrarily close to 2, see [Dav82,Sem86,Bis90]. Rather surprisingly, our Theorem 3.1 andTheorem 3.2 imply that Weil-Petersson quasicircles are closed under isometric welding.Moreover, IL(γ) ≤ IL(γ1) + IL(γ2).We describe this result more precisely in the case when both γ1 and γ2 are Jordan curvesthrough ∞ with finite energy (see [VW20a, Sec. 3.2] for the bounded curve case). LetHi, H

∗i be the connected components of Cr γi.

Corollary 3.9 ([VW20a, Cor. 3.4]). Let γ (resp. γ̃) be the arclength isometric weldingcurve of the domains H1 and H∗2 (resp. H2 and H∗1 ). Then γ and γ̃ have finite energy.Moreover,

IL(γ) + IL(γ̃) ≤ IL(γ1) + IL(γ2).

Proof. For i = 1, 2, let fi be a conformal equivalence H → Hi, and hi : H∗ → H∗i bothfixing ∞. By (2.4),

IL(γi) = DH(log |f ′i |

)+DH∗

(log |h′i|

).

Set ui := log |f ′i |, vi := log |h′i|. Then γ is the welding curve obtained from Theorem 3.2with u = u1, v = v2 and γ̃ is the welding curve for u = u2, v = v1. Then (3.4) implies

IL(γ) + IL(γ̃) ≤ DH (u1) +DH∗ (v2) +DH (u2) +DH∗ (v1) = IL(γ1) + IL(γ2)

as claimed.

The flow-line identity has the following corollary that we omit the proof. When γ is abounded finite energy Jordan curve (resp. finite energy Jordan curve passing through ∞),we let f be a conformal map from D (resp. H) to one connected component of Cr γ.

18

Corollary 3.10 ([VW20a, Cor. 1.5]). Consider the family of analytic curves γr := f(rT),where 0 < r < 1 (resp. γr := f(R + ir), where r > 0). For all 0 < s < r < 1 (resp.0 < r < s), we have

IL(γs) ≤ IL(γr) ≤ IL(γ), (resp. IL(γs) ≤ IL(γr) ≤ IL(γ), )

and equalities hold if only if γ is a circle (resp. γ is a line). Moreover, IL(γr) (resp. IL(γr))is continuous in r and

IL(γr)r→1−−−−−→ IL(γ); IL(γr)

r→0+−−−−→ 0

(resp. IL(γr) r→0+−−−−→ IL(γ); IL(γr) r→∞−−−→ 0).

Remark 3.11. Both limits and the monotonicity is consistent with the fact that theLoewner energy measures the “roundness” of a Jordan curve. In particular, the vanishingof the energy of γr as r → 0 expresses the fact that conformal maps asymptotically takesmall circles to circles.

4 Large deviations of multichordal SLE0+

4.1 Multichordal SLE

We now consider the multichordal SLEκ, that are families of random curves (multichords)connecting pairwise distinct boundary points of some planar domain. Constructions formultichordal SLEs have been obtained by many groups [Car03,Wer04a,BBK05,Dub07,KL07,Law09,MS16a,MS16b,BPW18,PW19], and models the interfaces in 2D statisticalmechanics models with alternating boundary condition.As in the single-chord case, we include the marked boundary points to the domain data(D;x1, . . . , x2n), assuming that they appear in counterclockwise order along the boundary∂D. The objects will be defined in a conformally invariant or covariant way, so withoutloss of generality, we assume that ∂D is smooth in a neighborhood of the marked points.Due to the planarity, there exist Cn different possible pairwise non-crossing connections forthe curves, where

Cn =1

n+ 1

(2nn

)(4.1)

is the n:th Catalan number. We enumerate them in terms of n-link patterns

α = {{a1, b1}, {a2, b2}, . . . , {an, bn}}, (4.2)

that is, partitions of {1, 2, . . . , 2n} giving a non-crossing pairing of the marked points.Now, for each n ≥ 1 and n-link pattern α, we let Xα(D;x1, . . . , x2n) ⊂

∏j X (D;xaj , xbj )

denote the set of multichords γ = (γ1, . . . , γn) consisting of pairwise disjoint chords whereγj ∈ X (D;xaj , xbj ) for each j ∈ {1, . . . , n}. We endow Xα(D;x1, . . . , x2n) with the relativeproduct topology and recall that X (D;xaj , xbj ) is endowed with the relative topologyinduced from a Hausdorff metric as in Section 2. Multichordal SLEκ is a random multichordγ = (γ1, . . . , γn) in Xα(D;x1, . . . , x2n), characterized in two equivalent ways, when κ > 0.

19

By re-sampling property: From the statistical mechanics model viewpoint, the naturaldefinition of multichordal SLE is such that for each j, the law of the random curve γj isthe chordal SLEκ in D̂j , conditioned on the other curves {γi | i 6= j}. Here, D̂j is thecomponent of D r

⋃i 6=j γi containing γj , highlighted in grey in Figure 2. In [BPW18], the

authors proved that multichordal SLEκ is the unique stationary measure of a Markov chainon Xα(D;x1, . . . , x2n) defined by re-sampling the curves from their conditional laws. Thisidea was already introduced and used earlier in [MS16a,MS16b], where Miller & Sheffieldstudied interacting SLE curves coupled with the Gaussian free field (GFF) in the frameworkof “imaginary geometry”.

x1

x2

x2n

xaj

xbjγj

Figure 2: Illustration of a multichord and the domain D̂j containing γj .

By Radon-Nikodym derivative: We assume1 that 0 < κ < 8/3. Multichordal SLE canbe obtained by weighting n independent SLEκ by

exp(c(κ)

2 mD(γ1, . . . , γn)), where c(κ) := (3κ− 8)(6− κ)2κ < 0, (4.3)

is the central charge associated to SLEκ. The quantity mD(γ) is defined using the Brownianloop measure µloopD introduced by Lawler, Schramm, and Werner [LSW03,LW04]:

mD(γ) :=n∑p=2

µloopD

({`∣∣ ` ∩ γi 6= ∅ for at least p chords γi})

=∫

max(#{chords hit by `} − 1, 0

)dµloopD (`)

which is positive and finite whenever the family (γi)i=1...n is disjoint. In fact, the Brownianloop measure is an infinite measure on Brownian loops, that is conformally invariant, andfor D′ ⊂ D, µloopD′ is simply µ

loopD restricted to loops contained in D′. When D has non-polar

boundary, the infinity of total mass of µloopD comes only from the contribution of smallloops. In particular, the summand {`

∣∣ ` ∩ γi 6= ∅ for at least p chords γi} is finite if p ≥ 2and the chords are disjoint. For n independent chordal SLEs connecting (x1, . . . , x2n), theymay intersect each other. However, in this case mD is infinite and the Radon-Nikodymderivative (4.3) vanishes since c < 0. We point out that mD(γ) = 0 if n = 1 (which is notsurprising since no weighting is needed for the single SLE).

1The same result holds for 8/3 ≤ κ ≤ 4, if one includes into the exponent the indicator function of theevent that all γj are pairwise disjoint.

20

Remark 4.1. Notice that when κ = 0, c = −∞. The second characterization does notapply and we show its existence and uniqueness by making links to rational functions inthe next section.

4.2 Real rational functions and Shapiro’s conjecture

From the re-sampling property, the multichordal SLE0 in Xα(D;x1, . . . , x2n) as a determin-istic multichord η = (η1, . . . , ηn) with the property that each ηj is the SLE0 curve in its owncomponent (D̂j ;xaj , xbj ). In other words, each ηj is the hyperbolic geodesic in (D̂j ;xaj , xbj ),see Remark 1.9. We call a multichord with this property a geodesic multichord. By theconformal invariance of the geodesic property, we assume that D = H without loss ofgenerality.The existence of geodesic multichord for each α follows by characterizing them as minimizersof a lower semicontinuous Loewner energy which is the large deviation rate function ofmultichordal SLE0+, to be discussed in the next section. Assuming the existence, theuniqueness is a consequence of the following algebraic result.

Theorem 4.2 ([PW20, Thm. 1.2, Prop. 4.1]). If η̄ ∈ Xα(H;x1, . . . , x2n) is a geodesicmultichord. The union of η̄, its complex conjugate, and the real line is the real locus of arational function hη of degree n+ 1 with critical points {x1, . . . , x2n}. The rational functionis unique up to post-composition by PSL(2,R) and by the map H→ H∗ : z 7→ −z.

A rational function is an analytic branched covering h of Ĉ over Ĉ, or equivalently, theratio of two polynomials. The degree of h is the number of preimages of any regular value.A point x0 ∈ Ĉ is a critical point (equivalently, a branched point) with index k (wherek ≥ 2) if

h(x) = h(x0) + C(x− x0)k +O((x− x0)k+1)

for some constant C 6= 0. In other words, the function is locally a k-to-1 branched cover inthe neighborhood of x0. By the Riemann-Hurwitz formula, a rational function of degreen + 1 has 2n critical points if and only if all of its critical points are of index two. Thegroup

PSL(2,R) ={A =

(a b

c d

): a, b, c, d ∈ R, ad− bc = 1

}/A∼−A

acts on H by A : z 7→ az+bcz+d , which is also called a Möbius transformation of H.We prove Theorem 4.2 by constructing the rational function associated to a geodesicmultichord η.

Proof. The complement Hrη has n+1 components that we call faces. We pick an arbitraryface F and consider a uniformizing conformal map hη from F onto H. Without loss ofgenerality, we consider the former case and assume that F is adjacent to η1. We call F ′ theother face adjacent to η1. Since η1 is a hyperbolic geodesic in D̂1, the map hη extends byreflection to a conformal map on D̂1. In particular, this extension of hη maps F ′ conformallyonto H∗. By iterating the analytic continuation across all of the chords ηk, we obtain a

21

meromorphic function hη : H→ Ĉ. Furthermore, hη also extends to H, and its restrictionhη|R takes values in R. Hence, Schwarz reflection allows us to extend hη to Ĉ by settinghη(z) := hη(z∗)∗ for all z ∈ H∗.Now, it follows from the construction that hη is a rational function of degree n + 1, asexactly n+ 1 faces are mapped to H and n+ 1 faces to H∗. Moreover, h−1η (R ∪ {∞}) isprecisely the union of η, its complex conjugate and R∪ {∞}. Finally, another choice of theface F we started with yields the same function up to post-composition by PSL(2,R) andz 7→ −z. This concludes the proof.

To find out all the geodesic multichords connecting {x1, . . . , x2n}, it thus suffices to classifyall the rational functions with this set of critical points. The following result is due toGoldberg.

Theorem 4.3 ([Gol91]). Let z1, . . . , z2n be 2n distinct complex numbers. There are atmost Cn rational functions (up to post-composition by a Möbius map of Ĉ in PSL(2,C)) ofdegree n+ 1 with critical points z1, . . . , z2n.

Assuming the existence of geodesic multichord in Xα(H;x1, . . . , x2n) and observing thattwo rational functions constructed in Theorem 4.2 are PSL(2,C) equivalent if and only ifthey are span〈PSL(2,R), z 7→ −z〉 equivalent, we obtain:

Corollary 4.4. There exists a unique geodesic multichord in Xα(D;x1, . . . , x2n) for each α.

The multichordal SLE0 is therefore well-defined. We also obtain a by-product of this result:

Corollary 4.5. If all critical points of a rational function are real, then it is a real rationalfunction up to post-composition by a Möbius transformation of Ĉ.

This corollary is a special case of the Shapiro conjecture concerning real solutions toenumerative geometric problems on the Grassmannian, see [Sot00]. In [EG02], Eremenkoand Gabrielov first proved this conjecture for the Grassmannian of 2-planes, when theconjecture is equivalent to Corollary 4.5. See also [EG11] for another elementary proof.

4.3 Large deviations of multichordal SLE0+

We now introduce the Loewner potential and energy and discuss the large deviations ofmultichordal SLE0+.

Definition 4.6. Let γ := (γ1, . . . , γn). The Loewner potential of γ is given by

HD(γ) :=112

n∑j=1

ID(γj) +mD(γ)−14

n∑j=1

logPD;xaj ,xbj , (4.4)

where ID(γj) = ID,xaj ,xbj (γj) is the chordal Loewner energy of γj (Definition 2.2) andPD;x,y is the Poisson excursion kernel, defined via

PD;x,y := |ϕ′(x)||ϕ′(y)|PH;ϕ(x),ϕ(y), and PH;x,y := |y − x|−2,

and where ϕ : D → H is a conformal map such that ϕ(x), ϕ(y) ∈ R.

22

When n = 1,HD(γ) =

112ID(γ)−

14 logPD;a,b.

We denote the minimal potential by

MαD(x1, . . . , x2n) := infγHD(γ) ∈ (−∞,∞), (4.5)

with infimum taken over all multichords γ ∈ Xα(D;x1, . . . , x2n).One important property of the Loewner potential is that it satisfies the following cascaderelation reducing the number of chords by one, see [PW20, Lem. 3.7, Cor. 3.8].

Lemma 4.7 (Cascade relation [PW20, Lem. 3.7]). For each j ∈ {1, . . . , n}, we have

HD(γ) = HD̂j (γj) +HD(γ1, . . . , γj−1, γj+1 . . . , γn). (4.6)

In particular, any minimizer of HD in Xα(D;x1, . . . , x2n) is a geodesic multichord, andHD(γ)

since c(κ)/2 ∼ −12/κ. The density of multichordal SLE is thus given by

exp(c(κ)

2 mD(γ))∏

j exp(− ID(γ)κ

)Eind exp

(c(κ)

2 mD) ∼κ→0+ exp(−IαD(γ)

κ

).

4.4 Minimal potential

To define the energy IαD, one could have added to the potential HD an arbitrary constantthat depends only on the boundary data (x1, . . . , x2n;α), e.g., one may drop the Poissonkernel terms in HD which also alters the value of the minimal potential. The advantageof introducing the Loewner potential (4.4) is that it allows comparing the potential ofgeodesic multichords of different boundary data. This becomes interesting when n ≥ 2 asthe moduli space of the boundary data is non-trivial. In this section we discuss equationssatisfied by the minimal potential based on [PW20] and the more recent [AKM20].We first use Loewner’s equation to describe the geodesic multichord, whose driving functionscan be expressed in terms of the minimal potential. We describe the result when D = H.

Theorem 4.13 ([PW20, Prop. 1.6]). Let η be the geodesic multichord in Xα(H;x1, . . . , x2n).For each j ∈ {1, . . . , n}, the Loewner driving function W of the chord ηj from xaj to xbjand the evolution V it = gt(xi) of the other marked points satisfy the differential equations

∂tWt = −12∂ajMαH(V 1t , . . . , Vaj−1t ,Wt, V

aj+1t , . . . , V

2nt ), W0 = xaj ,

∂tVit =

2V it −Wt

, V i0 = xi, for i 6= aj ,(4.7)

for 0 ≤ t < T , where T is the lifetime of the solution and (gt)t∈[0,T ] is the Loewner flowgenerated by ηj.

Here again, the idea of SLE large deviations enables us to speculate the form of Loewnerdifferential equations of the geodesic multichord. In fact, for each n-link pattern α, oneassociates to the multichordal SLEκ a (pure) partition function Zα defined as

Zα(H;x1, . . . , x2n) :=( n∏j=1

PH;xaj ,xbj

)(6−κ)/2κ× Eindκ exp

(c(κ)

2 mD(γ)),

where Eindκ denotes the expectation with respect to the law of the independent SLEκmultichord γ. As a consequence of the large deviation principle, we obtain that

κ logZα(H;x1, . . . , x2n)κ→0+−→ −12MαH(x1, . . . , x2n). (4.8)

The marginal law of the chord γκj in the multichordal SLEκ in Xα(H;x1, . . . , x2n) is givenby the stochastic Loewner equation derived from Zα:

dWt =√κdBt + κ∂aj logZα

(V 1t , . . . , V

aj−1t ,Wt, V

aj+1t , . . . , V

2nt

)dt, W0 = xaj ,

dV it =2dt

V it −Wt, V i0 = xi, for i 6= aj .

See [PW19, Eq. (4.10)]). Replacing naively κ logZα by −12MαH, we obtain (4.7).

24

To prove Theorem 4.13 rigorously, we analyse the geodesic multichords and the minimalpotential directly and do not need to go through SLE, which might be tedious to controlthe errors when interchanging derivatives and limits.

Proof of Theorem 4.13. Let us illustrate the case when n = 1. For n ≥ 2, we use aninduction and a conformal restriction formula which compares the driving function of asingle chord under conformal maps. See [PW20, Sec. 4.2] for the proof.When n = 1, the minimal potential has an explicit formula:

MH(x1, x2) =12 log |x2 − x1| =⇒ ∂1MH(x1, x2) =

12(x1 − x2)

. (4.9)

The hyperbolic geodesic in (H;x1, x2) is the semi-circle η with endpoints x1 and x2. Wecheck directly that ∂tWt|t=0 = 6/(x2 − x1). Since hyperbolic geodesic is preserved underits own Loewner flow, i.e., gt(η[t,T ]) is the semi-circle with end points Wt and V 2t = gt(x2),we obtain

∂tWt =6

V 2t −Wt, W0 = x1,

∂tV2t =

2V 2t −Wt

, V 20 = x2.

By (4.9), this is exactly Equation (4.7).

Similarly, using the level two null-state Belavin-Polyakov-Zamolodchikov equation satisfiedby the SLE partition function:κ

2∂2xj +

∑i 6=j

(2

xi − xj∂xi −

(6− κ)/κ(xi − xj)2

)Zα = 0, j = 1, . . . , 2n, (4.10)the same idea prompts us to find the following result, see also [BBK05,AKM20].

Theorem 4.14 ([PW20, Prop. 1.7]). Let Uα := −12MαH. For j ∈ {1, . . . , 2n}, we have12(∂jUα(x1, . . . , x2n))

2 +∑i 6=j

2xi − xj

∂iUα(x1, . . . , x2n) =∑i 6=j

6(xi − xj)2

. (4.11)

The recent work [AKM20] proves further an explicit expression of Uα(x1, · · · , x2n) in termsof the rational function hη associated to the geodesic multichord in Xα(H;x1, . . . , x2n) asconsidered in Section 4.2. More precisely, following [AKM20], we normalize the rationalfunction such that hη(∞) =∞ by possibly post-composing hη by an element of PSL(2,R)and denote the other n poles (ζα,1, · · · , ζα,n) of hη.

Theorem 4.15 ([AKM20]). For the boundary data (x1, . . . , x2n;α), we have

exp Uα = C∏

1≤j

5 Large deviations of radial SLE∞

We now turn to the large deviations of SLE∞, namely, when ε := 1/κ using the notation inDefinition 1.2. From (1.6), one can easily show that in the chordal setup, for any fixed t,the conformal map ft = g−1t : H→ HrKt converges uniformly on compacts to the identitymap as κ→∞ almost surely. In other words, the complement of the SLEκ hull convergesfor the Carathéodory topology towards H which is not interesting for the large deviations.The main hurdle is that the driving function can be arbitrarily close to the boundary point(i.e. ∞) where we normalize the conformal maps in the Loewner evolution. We thereforeswitch to the radial version of SLE.

5.1 Radial SLE

We now describe the radial SLE on the unit disk D targeting at 0. The radial Loewnerdifferential equation driven by a continuous function R+ → S1 : t 7→ ζt is defined as follows:for all z ∈ D,

∂tgt(z) = gt(z)ζt + gt(z)ζt − gt(z)

. (5.1)

As in the chordal case, the solution t 7→ gt(z) to (5.1) is defined up to the swallowing time

τ(z) := sup{t ≥ 0: infs∈[0,t]

|gs(z)− ζs| > 0},

and the growing hulls are given by Kt = {z ∈ D : τ(z) ≤ t}. The solution gt is the conformalmap from Dt := DrKt onto D satisfying gt(0) = 0 and g′t(0) = et.Radial SLEκ is the curve γκ tracing out the growing family of hulls (Kt)t≥0 driven by aBrownian motion on the unit circle S1 = {ζ ∈ C : |ζ| = 1} of variance κ, i.e.,

ζt := βκt = eiBκt , (5.2)

where Bt is a standard one dimensional Brownian motion. Radial SLEs exhibit the samephase transitions as in the chordal case as κ varies. In particular, when κ ≥ 8, γκ is almostsurely space-filling and Kt = γκ[0,t].We now argue heuristically to give an intuition about our results on the κ→∞ limit andlarge deviations in [APW20]. During a short time interval [t, t + ∆t] where the flow iswell-defined for a given point z ∈ D, we have gs(z) ≈ gt(z) for s ∈ [t, t+ ∆t]. Hence, writingthe time-dependent vector field (z(ζt + z)(ζt − z)−1)t≥0 generating the Loewner chain as(∫S1 z(ζ + z)(ζ − z)−1δβκt (dζ))t≥0, where δβκt is the Dirac measure at β

κt , we obtain that

∆gt(z) is approximately∫ t+∆tt

∫S1gt(z)

ζ + gt(z)ζ − gt(z)

δβκs (dζ)ds =∫S1gt(z)

ζ + gt(z)ζ − gt(z)

d(`κt+∆t(ζ)− `κt (ζ)), (5.3)

where `κt is the occupation measure (or local time) on S1 of βκ up to time t. As κ→∞,the occupation measure of βκ during [t, t+ ∆t] converges to the uniform measure on S1 oftotal mass ∆t. Hence the radial Loewner chain converges to a measure-driven Loewner

26

chain (also called Loewner-Kufarev chain) with the uniform probability measure on S1 asdriving measure, i.e.,

∂tgt(z) =1

2π

∫S1gt(z)

ζ + gt(z)ζ − gt(z)

|dζ| = gt(z).

This implies gt(z) = etz. Similarly, (5.3) suggests that the large deviations of SLE∞ canalso be obtained from the large deviations of the process (`κt )t≥0.

5.2 Loewner-Kufarev equations

The heuristic outlined above leads naturally to consider the Loewner-Kufarev chain drivenby measures that we now define more precisely. LetM(Ω) (resp. M1(Ω)) be the space ofBorel measures (resp. probability measures) on Ω. We define

N+ = {ρ ∈M(S1 × R+) : ρ(S1 × I) = |I| for all intervals I ⊂ R+}.

From the disintegration theorem (see e.g. [Bil95, Theorem 33.3]), for each measure ρ ∈ N+there exists a Borel measurable map t 7→ ρt from R+ toM1(S1) such that dρ = ρt(dζ) dt.We say (ρt)t≥0 is a disintegration of ρ; it is unique in the sense that any two disintegrations(ρt)t≥0, (ρ̃t)t≥0 of ρ must satisfy ρt = ρ̃t for a.e. t. We denote by (ρt)t≥0 one suchdisintegration of ρ ∈ N+.For z ∈ D, consider the Loewner-Kufarev ODE

∂tgt(z) = gt(z)∫S1

ζ + gt(z)ζ − gt(z)

ρt(dζ), g0(z) = z. (5.4)

Let τ(z) be the supremum of all t such that the solution is well-defined up to time t withgt(z) ∈ D, and Dt := {z ∈ D : τ(z) > t} is a simply connected open set containing 0. Thefunction gt is the unique conformal map of Dt onto D such that gt(0) = 0 and g′t(0) > 0.Moreover, it is straightforward to check that ∂t log g′t(0) = |ρt| = 1. Hence, g′t(0) = et,namely, Dt has conformal radius e−t seen from 0. We call (gt)t≥0 the Loewner-Kufarevchain driven by ρ ∈ N+.It is also convenient to use its inverse (ft := g−1t )t≥0, which satisfies the Loewner PDE :

∂tft(z) = −zf ′t(z)∫S1

ζ + zζ − z

ρt(dζ), f0(z) = z. (5.5)

We write L+ for the set of Loewner-Kufarev chains defined for time R+. An element of L+can be equivalently represented by (ft)t≥0 or (gt)t≥0 or the evolution family of domains(Dt)t≥0 or the evolution family of hulls (Kt = DrDt)t≥0.

Remark 5.1. In terms of the domains (Dt), according to a theorem of Pommerenke [Pom65,Satz 4] (see also [Pom75, Thm. 6.2] and [RR94]), L+ consists exactly of those (Dt)t≥0 suchthat Dt ⊂ D has conformal radius e−t and for all 0 ≤ s ≤ t, Dt ⊂ Ds.

We now restrict the Loewner-Kufarev chains to the time interval [0, 1] for the topology andsimplicity of notation. The results can be easily generalized to other finite intervals [0, T ]or to R+ as the projective limit of chains on all finite intervals. Define

N[0,1] = {ρ ∈M1(S1 × [0, 1]) : ρ(S1 × I) = |I| for all intervals I ⊂ [0, 1]},

27

endowed with the Prokhorov topology (the topology of weak convergence) and the corre-sponding set of restricted Loewner chains L[0,1]. Identifying an element (ft)t∈[0,1] of L[0,1]with the function f defined by f(z, t) = ft(z) and endow L[0,1] with the topology of uniformconvergence of f on compact subsets of D× [0, 1]. (Or equivalently, viewing L[0,1] as theset of processes (Dt)t∈[0,1], this is the topology of uniform Carathéodory convergence.) Thefollowing result allows us to study the limit and large deviations with respect to the topologyof uniform Carathéodory convergence.

Theorem 5.2 ([MS16d, Prop. 6.1], [JVST12]). The Loewner transform N[0,1] → L[0,1] :ρ 7→ f is a homeomorphism.

By showing that the random measure δβκt (dζ) dt ∈ N[0,1] converges almost surely to theuniform measure (2π)−1|dζ|dt on S1 × [0, 1] as κ→∞, we obtain:

Theorem 5.3 ([APW20, Prop. 1.1]). As κ→∞, the complement of hulls (Dt)t∈[0,1] of theradial SLEκ converges almost surely to (e−tD)t∈[0,1] for the uniform Carathéodory topology.

5.3 Loewner-Kufarev energy and large deviations

From the contraction principle Theorem 1.5 and Theorem 5.2, the large deviation principleof radial SLEκ as κ→∞ boils down to the large deviation principle of δβκt (dζ) dt ∈ N[0,1]for the Prokhorov topology. For this, we approximate δβκt (dζ) dt by

ρκn :=2n−1∑i=0

µκn,i(dζ)1t∈[i/2n,(i+1)/2n) dt,

where µκn,i ∈M1(S1) is the average of the measure δβκt on the time interval [i/2n, (i+1)/2n):

µκn,i = 2n(`κ(i+1)/2n − `κi/2n).

We start with the large deviation principle for µκn,i as κ → ∞. Let `κt := t−1`κt be the

average occupation measure of βκ up to time t. From the Markov property of Brownianmotion, we have µκn,i = `

κ2−n in distribution up to a rotation (by βκi/2n). The following result

is a special case of a theorem of Donsker and Varadhan.Define the functional IDV :M(S1)→ [0,∞] by

IDV (µ) = 12

∫S1|v′(ζ)|2 |dζ|, (5.6)

if µ = v2(ζ)|dζ| for some function v ∈W 1,2(S1) and ∞ otherwise.

Remark 5.4. Note that IDV is rotation-invariant and IDV (cµi) = cIDV (µi) for c > 0.

Theorem 5.5 ([DV75, Thm. 3]). Fix t > 0. The average occupation measure {`κt }κ>0admits a large deviation principle as κ→∞ with good rate function tIDV . Moreover, IDVis convex.

28

The above κ→∞ large deviation principle is in the sense of Definition 1.2 with ε = 1/κ,i.e., for any open set O and closed set F ⊂M1(S1),

limκ→∞

1κ

logP[`κt ∈ O] ≥ − infµ∈O

tIDV (µ);

limκ→∞

1κ

logP[`κt ∈ F ] ≤ − infµ∈F

tIDV (µ).

Theorem 5.5 implies that the 2n-tuple (µκn,0, . . . .µκn,2n−1) satisfies the large deviation prin-ciple with rate function

IDVn (µ0, . . . , µ2n−1) := 2−n2n−1∑i=0

IDV (µi).

Taking the n→∞ limit, we obtain the following definition.

Definition 5.6. We define the Loewner-Kufarev energy on L[0,1] (or equivalently on N[0,1])

S[0,1]((Dt)t∈[0,1]) := S[0,1](ρ) :=∫ 1

0IDV (ρt) dt

where ρ is the driving measure generating (Dt) and IDV .

Theorem 5.7 ([APW20, Thm. 1.2]). The measure δβκt (dζ) dt ∈ N[0,1] satisfies the largedeviation principle with good rate function S[0,1] as κ→∞.

Remark 5.8. Here we note that if ρ ∈ N[0,1] has finite Loewner-Kufarev energy, thenρt is absolutely continuous with respect to the Lebesgue measure for a.e. t with densitybeing the square of a function in W 1,2(S1). In particular, ρt is much more regular than aDirac measure. We see once more the regularizing phenomenon from the large deviationconsideration.

From the contraction principle Theorem 1.5 and Theorem 5.2, we obtain immediately:

Corollary 5.9 ([APW20, Cor. 1.3]). The family of SLEκ on the time interval [0, 1] satisfiesthe κ→∞ large deviation principle with the good rate function S[0,1].

6 Foliation of Weil-Petersson quasicircles

SLE processes enjoy a remarkable duality [Dub09a,Zha08a,MS16a] coupling SLEκ to theouter boundary of SLE16/κ for κ < 4. It suggests that the rate functions of SLE0+ (Loewnerenergy) and SLE∞ (Loewner-Kufarev energy) are also dual to each other. Let us firstremark that when S[0,1](ρ) = 0, the generated family (Dt)t∈[0,1] consists of concentric disks.In particular (∂Dt)t∈[0,1] are circles and thus have zero Loewner energy. This trivial examplesupports the guess that some form of energy duality holds.Viklund and I [VW20b] investigated the duality between these two energies, and moregenerally, the interplay with the Dirichlet energy of a so-called winding function. We nowdescribe briefly those results. While our approach is originally inspired by SLE theory,they are of independent interest from the analysis perspective and the proofs are alsodeterministic.

29

6.1 Whole-plane Loewner evolution

To describe our results in the most generality, we consider the Loewner-Kufarev energy forLoewner evolutions defined for t ∈ R, namely the whole-plane Loewner chain. We define inthis case the space of driving measures to be

N := {ρ ∈M(S1 × R) : ρ(S1 × I) = |I| for all intervals I}.

The whole-plane Loewner chain driven by ρ ∈ N , or equivalently by its measurable familyof disintegration measures R→M1(S1) : t 7→ ρt, is the unique family of conformal maps(ft : D→ Dt)t∈R such that

(i) For all s < t, 0 ∈ Dt ⊂ Ds.(ii) For all t ∈ R, ft(0) = 0 and f ′t(0) = e−t (i.e., Dt has conformal radius e−t).(iii) For all s ∈ R, (f (s)t := f−1s ◦ ft : D→ D

(s)t )t≥s is the Loewner chain driven by (ρt)t≥s,

which satisfies (5.5) with the initial condition f (s)s (z) = z as in Section 5.2.

See, e.g., [VW20b, Sec. 7.1] for a proof of the existence and uniqueness of such family.

Remark 6.1. If ρt is the uniform probability measure for all t ≤ 0, then ft(z) = e−tz fort ≤ 0 and (ft)t≥0 is the Loewner chain driven by (ρt)t≥0 ∈ N+. Indeed, we check directlythat (ft)t∈R satisfy the three conditions above. The Loewner chain in D is therefore aspecial case of the whole-plane Loewner chain.

Note that the condition (iii) is equivalent to for all t ∈ R and z ∈ D,

∂tft(z) = −zf ′t(z)∫S1

ζ + zζ − z

ρt(dζ). (6.1)

We remark that for ρ ∈ N , then ∪t∈RDt = C. Indeed, Dt has conformal radius e−t,therefore contains the centered ball of radius e−t/4 by Koebe’s 1/4 theorem. Since {t ∈ R :z ∈ Dt} 6= ∅, we define for all z ∈ C,

τ(z) := sup{t ∈ R : z ∈ Dt} ∈ (−∞,∞].

We say that ρ ∈ N generates a foliation (γt := ∂Dt)t∈R of Cr {0} if

1. For all t ∈ R, γt is a chord-arc Jordan curve (see Remark 2.17).2. It is possible to parametrize each curve γt by S1 so that the mapping t 7→ γt is

continuous in the supremum norm.3. For all z ∈ Cr {0}, τ(z)

0z

ϕ(z)

Figure 3: Illustration of the winding function ϕ.

6.2 Energy duality

The following result gives a qualitative relation between finite Loewner-Kufarev energymeasures and finite Loewner energy curves (i.e., Weil-Petersson quasicircle by Theorem 2.14).

Proposition 6.3 (Weil-Petersson leaves [VW20b, Thm. 1.1]). Suppose ρ is a measure onS1 × R with finite Loewner-Kufarev energy. Then ρ generates a foliation (γt = ∂Dt)t∈R ofCr {0} in which all leaves are Weil-Petersson quasicircles.

Every ρ with S(ρ) < ∞ thus generates a foliation of Weil-Petersson quasicircles thatcontinuously sweep out the Riemann sphere, starting from ∞ and moving towards 0, as tranges from −∞ to ∞. This class of measures seems to be a rare case for which a completedescription of the geometry of the generated non-smooth interfaces is possible.We also show a quantitative relation among energies by introducing a real-valued windingfunction ϕ associated to a foliation (γt)t∈R as follows. Let gt = f−1t : Dt → D. Since γt isby assumption chord-arc, thus rectifiable, γt has a tangent arclength-a.e. Given z at whichγt has a tangent, we define

ϕ(z) = limw→z

argwg′t(w)gt(w)

(6.2)

where the limit is taken inside Dt and approaching z non-tangentially. We choose thecontinuous branch of arg which vanishes at 0. Monotonicity of (Dt)t∈R implies that there isno ambiguity in the definition of ϕ(z) if z ∈ γt ∩ γs. See [VW20b, Sec. 2.3] for more details.

Remark 6.4. Geometrically, ϕ(z) equals the difference of the argument of the tangent ofγt at z and that of the tangent to the circle centered at 0 passing through z modulo 2π,see Figure 3.

In the trivial example when the measure ρ has zero energy, the generated foliation consistsof concentric circles centered at 0 whose winding function is identically 0. The following isour main theorem.

Theorem 6.5 (Energy duality [VW20b, Thm. 1.2]). Assume that ρ ∈ N generates afoliation and let ϕ be the associated winding function on C. Then DC(ϕ)

Remark 6.6. We have speculated the stochastic counterpart of this theorem in [VW20b,Sec. 10], that we call radial mating of trees, analogous but different from the mating of treestheorem of Duplantier, Miller, and Sheffield [DMS14] (see also [GHS19] for a recent survey).We do not enter into further details as there is no substantial theorem at this point.

Remark 6.7. A subtle point in defining the winding function is that in the general caseof chord-arc foliation, a function defined arclength-a.e. on all leaves need not be definedLebesgue-a.e., see, e.g., [Mil97]. Thus to consider the Dirichlet energy of ϕ, we consider thefollowing extension to W 1,2loc . A function ϕ defined arclength-a.e. on all leaves of a foliation(γt = ∂Dt) is said to have an extension φ in W 1,2loc if for all t ∈ R, the Jonsson-Wallin traceφ|γt of φ on γt defined by the following limit of averages on balls B(z, r) = {w : |w−z| < r}

φ|γt(z) := limr→0+

∫B(z,r)

φ dA (6.3)

coincides with ϕ arclength-a.e on γt. We also show that if such extension exists then it isunique. The Dirichlet energy of ϕ in the statement of Theorem 6.5 is understood as theDirichlet energy of this extension.

Theorem 6.5 has several applications which show that the foliation of Weil-Peterssonquasicircles generated by ρ with finite Loewner-Kufarev energy exhibits several remarkablefeatures and symmetries.The first is the reversibility of the Loewner-Kufarev energy. Consider ρ ∈ N and thecorresponding family of domains (Dt)t∈R. Applying z 7→ 1/z to Ĉ r Dt, we obtain anevolution family of domains (D̃t)t∈R upon time-reversal and reparametrization, which maybe described by the Loewner equation with an associated driving measure ρ̃. While thereis no known simple description of ρ̃ in terms of ρ, energy duality implies remarkably thatthe Loewner-Kufarev energy is invariant under this transformation.

Theorem 6.8 (Energy reversibility [VW20b, Thm. 1.3]). We have S(ρ) = S(ρ̃).

Proof sketch. The map z 7→ 1/z is conformal and the family of circles centered at 0is preserved under this inversion. We obtain from the geometric interpretation of ϕ(Remark 6.4) that the winding function ϕ̃ associated to the foliation (∂D̃t) satisfies ϕ̃(1/z) =ϕ(z) (one needs to work a bit as Remark 6.4 shows that the equality holds modulo 2π a priori).Since Dirichlet energy is invariant under conformal mappings, we obtain DC(ϕ̃) = DC(ϕ)and conclude with Theorem 6.5.

Remark 6.9. It is not known whether whole-plane SLEκ for κ > 8 is reversible. (Forκ ≤ 8, reversibility was established in [Zha15,MS17].) Therefore Theorem 6.8 cannot bepredicted from the SLE point of view by considering the κ→∞ large deviations as we didfor the reversibility of chordal Loewner energy in Theorem 2.7. This result on the otherhand suggests that reversibility for whole-plane radial SLE might hold for large κ as well.

From Proposition 6.3, being a Weil-Petersson quasicircle (separating 0 from ∞) is anecessary condition to be a leaf in the foliation generated by a finite energy measure. Thenext result shows that this is also a sufficient condition and we can relate the Loewner

32

energy of a leaf with the Loewner-Kufarev energy of measures generating it as a leaf. Weobtain a new and quantitative characterization of Weil-Petersson quasicircles.Let γ be a Jordan curve separating 0 from ∞, f (resp. h) a conformal map from D (resp.D∗) to the bounded (resp. unbounded) component of Cr γ fixing 0 (resp. fixing ∞).

Theorem 6.10 (Characterization of Weil-Petersson quasicircles [VW20b, Thm. 1.4]). Thecurve γ is a Weil-Petersson quasicircle if and only if γ can be realized as a leaf in thefoliation generated by a measure ρ with S(ρ)

the finite energy/large deviation world. Although some results involving the finite energyobjects are interesting on their own from the analysis perspective, we choose to omit fromthe table those without an obvious stochastic counterpart such as results in Section 3.3.We hope that the readers are now convinced that the ideas around large deviations of SLEare great sources generating exciting results in the deterministic world. Vice versa, as finiteenergy objects are more regular and easier to handle, exploring their properties would alsoprovide a way to speculate new conjectures in random conformal geometry. Rather thanan end, we hope it to be the starting point of new development along those lines and thissurvey can serve as a guide.

SLE/GFF with κ� 1 Finite energySection 1, 2

Chordal SLEκ in (D,x, y) A chord γ with ID,x,y(γ)

Multichordal SLEκ pure partition function Minimal potentialMαDZα of link pattern α (Definition 4.6, Equation (4.8))Loewner evolution of a chord in Loewner evolution of a chord in geodesicmultichordal SLEκ multichord (Theorem 4.13)Level 2 null-state BPZ equation Semiclassical null-state equationsatisfied by Zα satisfied byMαH (Theorem 4.14)

Section 5, 6Loewner chain (Dt)t∈R driven by ρ ∈ N

Whole-plane radial SLE16/κ with Loewner-Kufarev energy S(ρ)

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