Large Deviations of Schramm-Loewner Evolutions: a Survey

Large deviations of Schramm-Loewner evolutions: A survey

Yilin Wang

Massachusetts Institute of Technology [email protected]

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 functions and applications to complex analysis. More precisely, we describe the large deviations of SLEκ when the κ parameter goes to zero in the chordal and multichorcal case and to inﬁnity in the radial case. The rate functions, namely Loewner energy and Loewner-Kufarev energy, are of interest from the geometric function theory viewpoint and closely related to the Weil-Petersson class of quasicircles and real rational functions.

1 Contents

1 Introduction 3 1.1 Large deviation principle ...... 4 1.2 Chordal Loewner chain ...... 6 1.3 Chordal SLE ...... 7

2 Large deviations of chordal SLE0+ 8 2.1 Chordal Loewner energy and large deviations ...... 8 2.2 Reversibility of Loewner energy ...... 9 2.3 Loop energy and Weil-Petersson quasicircles ...... 10

3 Cutting, welding, and ﬂow-lines 13 3.1 Cutting-welding identity ...... 13 3.2 Flow-line identity ...... 16 3.3 Applications ...... 18

4 Large deviations of multichordal SLE0+ 19 4.1 Multichordal SLE ...... 19 4.2 Real rational functions and Shapiro’s conjecture ...... 21

4.3 Large deviations of multichordal SLE0+ ...... 22 4.4 Minimal potential ...... 24

5 Large deviations of radial SLE∞ 26 5.1 Radial SLE ...... 26 5.2 Loewner-Kufarev equations ...... 27 5.3 Loewner-Kufarev energy and large deviations ...... 28

6 Foliation of Weil-Petersson quasicircles 29 6.1 Whole-plane Loewner evolution ...... 30 6.2 Energy duality ...... 31

7 Summary 33

2 1 Introduction

The Schramm-Loewner evolution (SLE) is a model for conformally invariant random fractal curves in the plane, introduced by Schramm [Sch00] by combining Loewner’s classical theory [Loe23] for the evolution of planar slit domains with stochastic analysis. Schramm’s SLE is a one-parameter family of probability measures on non-self-crossing curves, indexed by κ ≥ 0, and thus denoted by SLEκ. When κ > 0, these curves are fractal, and the parameter κ reflects the curve’s roughness. SLEs play a central role in the recent development of 2D random conformal geometry. For instance, they describe interfaces in conformally invariant systems arising from scaling limits of discrete models in statistical physics, 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 closely related 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 of probability measures on an exponential scale. The formalization of the general framework of large deviation was developed by (or commonly attributed to) Varadhan [Var66]. A great deal of mathematics has been developed since then. Large deviations estimates have proved 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 both the SLE and large deviation theory, only sufficient for considering the large deviations of SLE. We by no means attempt to give a thorough reference to the background of these two theories. Our approach focuses on how large deviation consideration propels to the discovery (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 considered in 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, they still capture the essence of many coupling results in random conformal geometry. As for the large deviation literature, these results also show new ways to apply this powerful machinery. We summarize and compare the quantities and theorems from both random conformal geometry and the finite energy world in the last section. Impatient readers may skip 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 the intuition behind the theorems and sometimes omit proofs or only present the proof in a simpler case to illustrate the idea. These lecture notes are written based on two lecture series that I gave at the joint webinar of Tsinghua-Peking-Beijing Normal Universities and Random Geometry and Statistical Physics online seminars during the Covid-19 time. I would like to thank the organizers for the invitation and the online lecturing experience under the pandemic’s unusual situation and is supported by the NSF grant DMS-1953945.

3 1.1 Large deviation principle

We consider ﬁrst a simple example to illustrate the concept of large deviations. Let X ∼ N (0, σ2) be a real, centered Gaussian random variable of variance σ2. The density function of X is given by 1 x2 pX (x) = √ exp(− ). 2πσ2 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

√ 1 Z ∞ x2 √ P( εX ≥ M) = exp(− 2 )dx. 2πσ2ε M 2σ ε To quantify how rare this event happens when ε → 0+, we have ! √ 1 Z ∞ x2 √ ε log P( εX ≥ M) = ε log exp(− 2 )dx 2πσ2ε M 2σ ε Z ∞ 2 1 2 x = − ε log(2πσ ε) + ε log exp(− 2 )dx 2 M 2σ ε 2 ε→0+ M −−−−→− =: −IX (M) = − inf IX (x). 2σ2 x∈[M,∞)

The function I : x 7→ x2/2σ2 is called the large deviation rate function of the family of X √ random variables ( εX)ε>0. Now let us state the large deviation principle more precisely. Let X be a Polish space, B its Borel σ-algebra, {µε}ε>0 a family of probability measures on (X , B). To reduce possible measurability 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 function is 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) satisﬁes the large deviation principle of rate function I if for all open sets O ∈ B and closed set F ∈ B, lim ε log µε(O) ≥ − inf I(x); lim ε log µε(F ) ≤ − inf I(x). ε→0+ x∈O ε→0+ x∈F

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 continuity set of I), then the large deviation principle gives

lim ε log µε(A) = − inf I(x). ε→0+ x∈A √ 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 .

4 The reader should mind carefully that large deviation results depend on the topology involved which can be a subtle point. On the other hand, it follows from the deﬁnition that 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 , satisfying the large deviation principle with good rate function I : X → [0, ∞]. Let I0 : Y → [0, ∞] be deﬁned as I0(y) := inf I(x). x∈θ−1{y}

Then the family of pushforward probability measures (θ∗µε)ε>0 on Y satisﬁes the large deviation principle with good rate function I0.

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

Z T 1 2 IT (W ) := |∂tWt| dt, (1.1) 2 0 if W is absolutely continuous, and set to equal ∞ otherwise. Equivalently, we can write

k−1 (W − W )2 I (W ) = sup X ti+1 ti (1.2) T 2(t − t ) Π i=0 i+1 i 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 linear interpolation 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 √ 0 {( εBt)t∈[0,T ]}ε>0 satisfies the following large deviation principle in (C [0,T ], k·k∞) 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, whereas Brownian motion is only a.s. (1/2 − δ)-Hölder for δ > 0. However, Schilder’s theorem shows that Brownian motion singles out the Dirichlet energy which quantifies, as ε → 0+, the density of Brownian path around a deterministic function W . In fact, let Oδ(W ) be the δ 0 neighborhood of W in C [0,T ]. From the monotonicity of inf ˜ I (W˜ ) in δ, O (W ) W ∈Oδ(W ) T δ is a continuity set for IT with exceptions for at most countably many δ. Hence, we have

√ ε→0+ δ→0+ −ε log ( εB ∈ Oδ(W )) −−−−→ inf IT (W˜ ) −−−−→ IT (W ). (1.3) P ˜ W ∈Oδ(W ) 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 the large 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

5 of independent Gaussian random variables (B − B ) with variances (t − t ) ti+1√ ti 0≤i≤k−1 i+1 i respectively. Multiplying the Gaussian vector by ε, we obtain the large deviation principle (W −W )2 of the finite dimensional marginal with rate function Pk−1 ti+1 ti from Remark 1.4, i=0 2(ti+1−ti) Theorem 1.5, and the independence of the family of increments. Approximating the Brownian motion on the finite interval [0,T ] by the linearly interpolated function from its value at times ti, it suggests that the scaled Brownian paths satisfy the large deviation principle of rate function the supremum of the rate function of all of its finite dimensional marginals 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 allows generalization to any abstract Wiener space. Namely, the associated family of Gaussian √ measures 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 deviation principle of projective limits shows that Schilder’s theorem also holds for the inﬁnite time Brownian motion, when C0[0, ∞) is endowed with the topology of uniform convergence on 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 that encodes a non-self-intersecting curve on a 2-D domain into a driving function. In this survey, we use two types of Loewner chain: the chordal Loewner chain in (D; x, y), where D is a simply connected domain with two distinct boundary points x (starting point) and y (target point); and later in Section 5, the radial Loewner chain in D targeting at an interior 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 the ∗ article, H = {z ∈ C : Im(z) > 0} is the upper halfplane, H = {z ∈ C : Im(z) < 0} is the ∗ lower 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 from 0 to ∞ in H. We parameterize the curve by the halfplane capacity. More precisely, γ is continuously parametrized by R+, with γ0 = 0, γt → ∞ as t → ∞, in the way such that the unique conformal map gt from H r γ[0,t] onto H with the expansion at infinity gt(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 extended by continuity to the boundary point γt and that the real-valued function Wt := gt(γt) is continuous. This function W is called the driving function of γ. Conversely, the chordal Loewner chain in (H; 0, ∞) driven by a continuous real-valued 0 function W ∈ C [0, ∞) is the family of conformal maps (gt)t≥0, obtained by solving the Loewner equation for each z ∈ H, 2 ∂tgt(z) = with initial condition g0(z) = z. (1.6) gt(z) − Wt

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

τ(z) := sup{t ≥ 0: inf |gs(z) − Ws| > 0} s∈[0,t] 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 H r Kt onto H with expansion z + o(1) as z → ∞. Moreover, Kt has halfplane capacity 2t. Clearly Kt 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 the solution 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+s r Ks) − 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 unknown how 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 strictly less than 4 [MR05, Lin05].

1.3 Chordal SLE

We now very brieﬂy review the deﬁnition and relevant properties of chordal SLE. For further SLE background, we refer the readers to, e.g., [Law05, Wer04b]. The chordal Schramm-Loewner evolution of parameter κ in ( ; 0, ∞), denoted by SLE , is the random H √ κ non-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 the interfaces 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 invariant systems is that they are the unique random Loewner chain that are scaling-invariant and satisfy the domain Markov property. More precisely, for λ > 0, the law is invariant under the scaling transformation (then reparametrize by capacity)

λ (Kt)t≥0 7→ (Kt := λKλ−2t)t≥0

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

7 In fact, these two properties on (Kt) translate into the independent stationary increments property (i.e., being a continuous Lévy process) and the invariance under Wt λWλ−2t of the driving function. Multiples of Brownian motions are the only continuous processes satisfying 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 taking respectively the boundary points x, y to 0, ∞ (another choice of ϕ˜ equals λϕ for some λ > 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 at 0. 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 goes to ∞ 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 n o [ [ dh(F1,F2) := inf ε ≥ 0 F1 ⊂ Bε(x) and F2 ⊂ Bε(x) , x∈F2 x∈F1 where Bε(x) denotes the Euclidean ball of radius ε centered at x ∈ D. We then define the Hausdorff metric on the set of closed subsets of D via the pullback by a conformal map D → D. Although the metric depends on the choice of the conformal map, the topology induced 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 unparametrized simple 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 Dirichlet energy (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 the driving 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 two choices diﬀer only by post-composing by a scaling factor. From the scaling property of the

8 Loewner driving function, W changes to t 7→ λWλ−2t which has the same Dirichlet energy as W . The Loewner energy ID;x,y(γ) is non-negative and minimized by the hyperbolic geodesic since 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 ) < ∞, then W generates a simple chord in S (H; 0, ∞), namely, γ = t≥0 Kt ∈ X (H; 0, ∞).

Theorem 1.6 suggests that the Loewner energy is the large deviation rate function of {SLEκ}κ>0, with ε = κ. Indeed, the following result is proved in [PW20] which strengthens a similar result in [Wan19a]. As we are interested in the 0+ limit, we only consider small κ κ where the trace γ := ∪t≥0Kt of SLEκ is almost surely in X (D; x, y).

κ Theorem 2.4 ([PW20, Thm. 1.4]). The family of distributions {P }κ>0 on X (D; x, y) of the 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

κ κ lim κ log P [γ ∈ O] ≥ − inf ID;x,y(γ), κ→0+ γ∈O κ κ lim κ log P [γ ∈ F ] ≤ − inf ID;x,y(γ), κ→0+ γ∈F 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 of hulls it generates, endowed with the Hausdorﬀ metric, is not continuous and the contraction principle (Theorem 1.5) does not apply. Therefore Schilder’s theorem does not imply trivially 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 regular than SLEκ curves for any κ > 0. In fact, as we will see in Theorem 2.14, finite energy chord is part of a Weil-Petersson quasicircle which is rectifiable, see Remark 2.17. On the other hand, 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 statistical physics, it was natural to conjecture that chordal SLE is reversible, i.e., the time-reversal of a 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 simple curves κ ∈ [0, 4] via rather non-trivial couplings of both ends of the SLE path. See also Dubédat’s commutation relations [Dub07], and Miller and Sheffield’s approach based on the Gaussian Free Field [MS16a,MS16b,MS16c] that also provides a proof in the non-simple curve case when κ ∈ (4, 8].

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

9 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 any chord γ ∈ X (D; x, y).

Proof. Without loss of generality, we assume that (D; x, y) = (H; 0, ∞). We want to show that IH;0,∞(γ) = IH;0,∞(ι(γ)).

Let δn be a sequence of numbers converging to 0+, such that Oδn (γ) = {γ˜ ∈ X (H; 0, ∞): θ∗dh(γ, γ˜) < δn} is a continuity set for IH;0,∞. The sequence exists since there are at most a countable number of exceptions of δ such that Oδ(γ) is not a continuity set. In fact, for 0 δ < δ , we have the inclusion Oδ(γ) ⊂ Oδ(γ) ⊂ Oδ0 (γ) and

inf IH;0,∞(˜γ) ≤ inf IH;0,∞(˜γ) ≤ inf IH;0,∞(˜γ). γ˜∈Oδ0 (γ) γ˜∈Oδ(γ) γ˜∈Oδ(γ)

Therefore the function infγ˜∈Oδ(γ) IH;0,∞(γ˜) has a jump at points δ > 0 for which Oδ(γ) is not a continuity set. The monotonicity of δ 7→ infγ˜∈Oδ(γ) IH;0,∞(γ˜) shows that we have at most countably many such discontinuity points. From Remark 1.3 and Theorem 2.6, we have κ κ lim κ log P (γ ∈ Oδn (γ)) = − inf IH;0,∞(˜γ), (2.2) κ→0+ γ˜∈Oδn (γ) 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 metric −1 in H as in Definition 2.1. This choice makes ϑ ◦ ι ◦ ϑ = − IdD. In other words, ι induces an isometry on closed sets of H. Theorem 2.6 then shows that

κ κ κ κ κ κ P (γ ∈ Oδn (γ)) = P (ι(γ ) ∈ ι(Oδn (γ))) = P (γ ∈ 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 the spirit. We used here Theorem 2.4 from the recent work [PW20], whereas the original proof in [Wan19a] used the more complicated left-right passing events without the strong large deviation result at hand.

Remark 2.9. The energy reversibility is a result about deterministic chords. However, from the 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 energy that is constant after time 1 (and contributes 0 energy). From the additivity property of driving function, γ[1,∞) is the hyperbolic geodesic in H r γ[0,1]. The reversed curve ι(γ) is a chord starting with an analytic curve which deviates from the imaginary axis. Therefore unlike 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 Riemann sphere Cˆ = C ∪ {∞}. This generalization will show that the Loewner energy exhibits more

10 γ C \ R+ z 7→ z2 0 H γ 0 ∞ 0

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

L I (γ, γ(0)) := lim Iˆ (γ[ε, 1]). ε→0 Crγ[0,ε],γ(ε),γ(0)

The loop energy generalizes the chordal energy. In fact, let η be a simple chord in (C r 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,

L I (γ, ∞) =I ,0,∞(η) + lim Iˆ (γ[ε, 1/2]) CrR+ ε→0 Crγ[0,ε],γ(ε),γ(0)

=ICrR+,0,∞(η), since γ[ε, 1/2] is contained in the hyperbolic geodesic between γ(ε) and γ(0) in Cˆ 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 from Theorem 2.14, see Remark 2.16.

Remark 2.11. From the deﬁnition, the loop energy IL is invariant under the Möbius L transformations of Cˆ, and I (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 Cˆ constructed in [Zha20] (see also [Wer08, BD16] for the construction when κ = 8/3 and 2). However, the conformal invariance of the SLE loop measures implies that they have inﬁnite total mass and has to be normalized properly for considering large deviations. 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 in Section 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 quasiconformal map 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 Hausdorff dimension larger than 1. The natural question is then to identify the family of finite energy quasicircles. The answer is surprisingly a family of so-called Weil-Petersson quasicircles, which has been studied by both physicists and mathematicians since the eighties, including Bowick, Rajeev, Witten, Nag, Verjovsky, Sullivan, Cui, Tahktajan, Teo, Shen, Sharon, Mumford, Pommerenke, González, Bishop, etc., and is still an active research area. See the introduction of the recent preprint [Bis19] for a summary and a list of (currently) more than twenty equivalent definitions of very different nature. We will use the following definition. The class of Weil-Petersson quasicircles is preserved under Möbius transformation, so without loss of generality, we only spell out the definition of a bounded Weil-Petersson quasicircle. Let γ be a bounded Jordan curve, and write Ω ∗ and Ω be the connected components of Cˆ r γ. Let f be a conformal map from D onto the ∗ bounded component Ω, and h a conformal map from D onto the unbounded component Ω∗ fixing ∞.

Deﬁnition 2.13. The bounded Jordan curve γ is a Weil-Petersson quasicircle if and only if the following equivalent conditions hold: 1. D (log |f 0|) := 1 R |∇ log |f 0| (z)|2 dA(z) = 1 R |f 00(z)/f 0(z)|2 dA(z) < ∞; D π D π D 0 2. DD∗ (log |h |) < ∞, where dA denotes the Euclidean area measure.

Theorem 2.14 ([Wan19b, Thm. 1.3]). A bounded Jordan curve γ has ﬁnite Loewner energy if and only if γ is a Weil-Petersson quasicircle. Moreover, we have the identity

L 0 0 0 0 I (γ) = DD(log f ) + DD∗ (log h ) + 4 log f (0) − 4 log h (∞) , (2.3)

0 0 where h (∞) := limz→∞ h (z). Remark 2.15. If γ is a Jordan curve passing through ∞, then

L 0 0 I (γ) = DH(log f ) + DH∗ (log h ) (2.4)

∗ ∗ where f and h map conformally H and H onto, respectively, H and H , the two components of C r γ, while ﬁxing ∞. See [Wan19b, Thm. 1.1]. In fact, the identity (2.4) is proved ﬁrst, by considering approximation of γ by curves generated by piecewise linear driving function, in which case (2.4) is checked by explicit computation. We then establish from (2.4) an expression of IL(γ) in terms of zeta- regularized determinants of Laplacians [Wan19b, Thm. 7.3] via the Polyakov-Alvarez formula. The expression using determinants has the advantage of being invariant under conformal change of metric and allows us to move the point ∞ away from γ and obtain (2.3).

Remark 2.16. Note that the proof of Theorem 2.14 outlined above is purely deterministic and does not rely on previous results on the reversibility and root-invariance of the Loewner energy. The right-hand side of (2.3) clearly does not depend on any parametrization of γ, thus this theorem provides another proof of Theorem 2.10 and Theorem 2.7.

0 0 Remark 2.17. Since DD(log |f |) < ∞, log f belongs to VMOA, the space of functions in the Hardy space H2 with vanishing mean oscillations. A theorem of Pommerenke [Pom78]

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

lim length (γx,y)/|x − y| = 1. |x−y|→0

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

The connection between Loewner energy and Weil-Petersson quasicircles goes further: Not only the Loewner energy identiﬁes Weil-Petersson quasicircles from its ﬁniteness, but is also 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-hand side of (2.3) coincides with the universal Liouville action introduced by Takhtajan and Teo [TT06] and shown by them to be a Kähler potential of the Weil-Petersson metric, which is 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 metric on T0(1).

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

3 Cutting, welding, and ﬂow-lines

Pioneering works [Dub09b,She16,MS16a] on couplings between SLEs and Gaussian free ﬁeld (GFF) have led to many remarkable applications in the study of 2D random conformal geometry. 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 of large deviations by showing the interplay between Loewner energy of curves and Dirichlet energy of functions deﬁned in the complex plane (which is the large deviation rate function of scaled GFF). These results are analogous to the SLE/GFF couplings, but the proofs are remarkably 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 with weak ﬁrst derivatives in L2(Ω) and recall the Dirichlet energy of ϕ ∈ E(Ω) is Z 1 2 DΩ(ϕ) := |∇ϕ| dA(z). π Ω Theorem 3.1 (Cutting [VW20a, Thm. 1.1]). Suppose γ is a Jordan curve through ∞ and ϕ ∈ E(C). Then we have the identity:

L DC(ϕ) + I (γ) = DH(u) + DH∗ (v), (3.1)

13 where 0 0 u = 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 components of C r γ, while ﬁxing ∞.

The function ϕ ∈ E(C) implies that it has vanishing mean oscillation. The John-Nirenberg inequality (see, e.g., [Gar07, Thm. VI.6.4]) shows that e2ϕ is locally integrable and strictly 2ϕ positive. In other words, e dA(z) defines a σ-finite measure supported on C, absolutely continuous with respect to Lebesgue measure dA(z). The transformation law (3.2) is chosen 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/GFF coupling. 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 surfaces via a cutting operation and as welding curves [She16, DMS14]. A quantum surface is a √ domain 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 energy L SLEκ loop Jordan curve γ with I (γ) < ∞ i.e., a Weil-Petersson quasicircle γ √ Free boundary GFF κΦ on (on ) 2u ∈ E( ) (2ϕ ∈ E( )) H √ C H C √ κΦ 2ϕ κ-LQG on quantum plane ≈ e dA(z) e dA(z), ϕ ∈ E(C) √ κ-LQG on quantum half-plane on e2u dA(z), u ∈ E( ) H√ H √ κΦ/2 u 1/2 κ-LQG boundary measure on R ≈ e dx e dx, u ∈ H (R) SLEκ cuts an independent A Weil-Petersson quasicircle γ cuts √ quantum plane e κΦdA(z) into ϕ ∈ E( ) into u ∈ E( ), v ∈ E( ∗), √ √ C H H κΦ1 κΦ2 L ind. quantum half-planes e , e I (γ) + DC(ϕ) = DH(u) + DH∗ (v)

To explain the analogy in the last line, we argue as follows. From the left-hand side, one expects that under appropriate topology and for small κ, √ “P(SLEκ loop stays close to γ, κΦ stays close to 2ϕ) √ √ = P( κΦ1 stays close to 2u, κΦ2 stays close to 2v)”.

From the large deviation principle and the independence between SLE and Φ, we obtain similarly as (1.4) √ “ lim −κ log (SLEκ stays close to γ, κΦ stays close to 2ϕ) κ→0 P √ = lim −κ log (SLEκ stays close to γ) + lim −κ log ( κΦ stays close to 2ϕ) κ→0 P κ→0 P L = I (γ) + DC(ϕ)”.

14 On the other hand the independence between Φ1 and Φ2 gives √ √ “ lim −κ log ( κΦ1 stays close to 2u, κΦ2 stays close to 2v) κ→0 P

= 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 and ∞ ϕ ∈ Cc (C) to illustrate the idea. The general case follows from an approximation argument, see [VW20a] for the complete proof.

Proof. From Remark 2.15, if γ passes through ∞, then Z Z L 1 2 1 2 I (γ) = |∇σf | dA(z) + |∇σh| dA(z), π H π H∗ 0 0 where σf and σh are the shorthand notation for log |f | and log |h |. The conformal invariance of 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 cross terms that Z Z h∇σf (z), ∇(ϕ ◦ f)(z)i dA(z) + h∇σh(z), ∇(ϕ ◦ h)(z)i dA(z) = 0. (3.3) H H∗ By Stokes’ formula, the first term on the left-hand side equals Z Z Z 0 ∂nσf (x)ϕ(f(x))dx = k(f(x)) f (x) ϕ(f(x))dx = k(z)ϕ(z) |dz| R R ∂H where k(z) is the geodesic curvature of γ = ∂H at z using the identity ∂nσf (x) = |f 0(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 as a point of ∂H∗, equals −k(z). Therefore the sum in (3.3) cancels out and completes the proof in the smooth case.

The following result is on the converse operation of the cutting, which shows that we can also recover γ and ϕ from u and v by conformal welding. More precisely, an increasing homeomorphism w : R → R is said to be a (conformal) welding homeomorphism of a Jordan curve γ through ∞, if there are conformal maps f, h of the upper and lower half-planes −1 onto the two components of C r γ, respectively, such that w = h ◦ f|R. Suppose H and ∗ H are each equipped with an infinite and positive boundary measure defining a metric on R such that the distance between x < y equals the measure of [x, y]. Under suitable ∗ assumptions, the increasing isometry w : R = ∂H → ∂H = R fixing 0 is well-defined and a welding 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) and ∗ u v v ∈ E(H ) are given. The isometric welding problem for the measures e dx and e dx on R has a solution (γ, f, h) and the welding curve γ has ﬁnite Loewner energy. Moreover, there exists a unique ϕ ∈ E(C) such that (3.2) is satisﬁed.

15 u v In the statement, the measures e dx and e dx are deﬁned using the traces of u, v on R, 1/2 1/2 that are H (R) functions. We say that u ∈ H (R) if ZZ 2 2 1 |u(x) − u(y)| kukH1/2 := 2 2 dx dy < ∞. 2π R×R |x − y|

1/2 u Moreover, as H (R) ⊂ VMO(R), John-Nirenberg inequality implies that e dx is a σ-ﬁnite measure supported on R.

Proof sketch. We prove first that the increasing isometry w : R → R obtained from the measures eudx and evdx satisfies log w0 ∈ H1/2, which is equivalent to w being the welding homeomorphism of a Weil-Petersson quasicircle [ST20, STW18] and shows the existence of solution (γ, f, h). We then check that the function ϕ defined a priori in C r γ from u, v and the transformation law (3.2), extends to a function in E(C). See [VW20a, Sec. 3.1] for more details.

Remark 3.3. The solution (γ, f, h) in Theorem 3.2 is unique if appropriately normalized since quasicircles are conformally removable. Theorem 3.1 then shows an upper bound of the welded curve’s energy:

L I (γ) = DH(u) + DH∗ (v) − DC(ϕ) ≤ DH(u) + DH∗ (v). (3.4)

3.2 Flow-line identity

Now let us turn to the second identity between Loewner energy and Dirichlet energy. Since the Dirichlet energy of a harmonic function equals that of its harmonic conjugate, (2.4) can be written as Z Z L 1 0 2 1 0 2 I (γ) = ∇ arg f dA(z) + ∇ arg h dA(z). π H π H∗ On the other hand, if γ has finite energy, the boundary value of (a continuous branch of) arg f 0 ◦ f −1 equals the winding τ = arg γ0 along the curve γ, and one can express the Loewner energy as L I (γ) = DC(P[τ]), (3.5) where we write P[τ] for the harmonic extension of τ to C r γ on both sides. 0 ˆ We interpret (3.5) in the following way. Let ϕ ∈ E(C) ∩ C (C), namely, EC(ϕ) < ∞, ϕ is iϕ(z) continuous and admits a limit in R as z → ∞.A flow-line of the unit vector e through a point z0 ∈ C is a solution to the differential equation

iϕ(γ(t)) γ˙ (t) = e , t ∈ (−∞, ∞), γ(0) = z0.

The solution always exists by the Cauchy-Peano theorem but may not be unique, see [VW20a, Thm. 3.10]. In particular, for all z ∈ γ, ϕ(z) = τ(z). The orthogonal decomposition of ϕ for the Dirichlet inner product gives ϕ = P[ϕ|γ] + ϕ0, where ϕ0 = ϕ − P[ϕ|γ] has zero trace on γ and P[τ] is harmonic in C r γ. Combining with (3.5) we obtain the following result.

16 0 Theorem 3.4 (Flow-line identity [VW20a, Thm. 1.4]). Let ϕ ∈ E(C) ∩ C (Cˆ). Any flow- line γ of the vector field eiϕ is a Jordan curve through ∞ with finite Loewner energy and we have the formula L DC(ϕ) = I (γ) + DC(ϕ0). (3.6)

0 Remark 3.5. The assumption of ϕ ∈ C (Cˆ) is for technical reason to consider the flow-line of eiϕ in the classical differential equation sense. One may also drop this assumption by defining a flow-line to be a chord-arc curve γ passing through ∞ on which ϕ = τ arclength almost everywhere. We will further explore these ideas in a setting adapted to bounded curves (see Theorem 6.13).

This identity is analogous to the flow-line coupling between SLE and GFF, of critical importance, e.g., in the imaginary geometry framework of Miller-Sheffield [MS16a]: very loosely speaking, an SLEκ curve may be coupled with a GFF Φ and thought of as a flow-line of 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 complex identity for a bounded Jordan curve.

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

DC(ψ) = DH(ζ) + DH∗ (ξ), where ζ = ψ ◦ f + (log f 0)∗, ξ = ψ ◦ h + (log h0)∗ 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 Im ψ, as the real part of ψ only contributes to a reparametrization of γ.

Proof. From the identity arg f 0 = P[Im ψ] ◦ f, we have

0 0 ζ = Re ψ ◦ f + log |f | + i Im ψ ◦ f − arg f = u + i Im ψ0 ◦ f;

ξ = v + i Im ψ0 ◦ h,

L DC(Re ψ) + I (γ) = DH(u) + DH∗ (v).

L On the other hand, the ﬂow-line identity gives DC(Im ψ) = I (γ) + DC(Im ψ0). Hence,

L DC(ψ) = DC(Re ψ) + DC(Im ψ) = DC(Re ψ) + I (γ) + DC(Im ψ0)

= DH(u) + DH∗ (v) + DC(Im ψ0)

= DH(ζ) + DH∗ (ξ) as claimed.

17 Remark 3.8. From Corollary 3.6 we can easily recover the ﬂow-line identity (Theorem 3.4), by taking Im ψ = ϕ and Re ψ = 0. Similarly, the cutting-welding identity (3.1) follows from taking Re ψ = ϕ and Im ψ = 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-line identities.

3.3 Applications

We now show that these identities between Loewner and Dirichlet energies inspired by probabilistic couplings, have interesting consequences in geometric function theory.

Suppose γ1, γ2 are locally rectifiable Jordan curves of the same length (possibly both infinite) bounding two domains Ω1 and Ω2 and we mark a point on each curve. Let w be an arclength isometry γ1 → γ2 matching the marked points. We obtain a topological sphere from Ω1 ∪ Ω2 by identifying the matched points. Following Bishop [Bis90], the isometric welding problem is to find a Jordan curve γ ⊂ Cˆ, and conformal equivalences ˆ −1 f1, f2 from Ω1 and Ω2 to the two connected components of C r γ, such that f2 ◦ 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 nor the uniqueness of γ, but the chord-arc property does. However, chord-arc curves are not closed under isometric conformal welding: the welding curve can have Hausdorff dimension arbitrarily close to 2, see [Dav82, Sem86, Bis90]. Rather surprisingly, our Theorem 3.1 and Theorem 3.2 imply that Weil-Petersson quasicircles are closed under isometric welding. L L L Moreover, I (γ) ≤ I (γ1) + I (γ2).

We describe this result more precisely in the case when both γ1 and γ2 are Jordan curves through ∞ with ﬁnite energy (see [VW20a, Sec. 3.2] for the bounded curve case). Let ∗ Hi,Hi be the connected components of C r γi. Corollary 3.9 ([VW20a, Cor. 3.4]). Let γ (resp. γ˜) be the arclength isometric welding ∗ ∗ curve of the domains H1 and H2 (resp. H2 and H1 ). Then γ and γ˜ have ﬁnite energy. Moreover, L L L L I (γ) + I (˜γ) ≤ I (γ1) + I (γ2).

∗ ∗ Proof. For i = 1, 2, let fi be a conformal equivalence H → Hi, and hi : H → Hi both ﬁxing ∞. By (2.4), L 0 0 I (γi) = DH log |fi | + DH∗ log |hi| .

0 0 Set ui := log |fi |, vi := log |hi|. Then γ is the welding curve obtained from Theorem 3.2 with u = u1, v = v2 and γ˜ is the welding curve for u = u2, v = v1. Then (3.4) implies

L L L L I (γ) + I (˜γ) ≤ DH (u1) + DH∗ (v2) + DH (u2) + DH∗ (v1) = I (γ1) + I (γ2) as claimed.

The flow-line identity has the following corollary that we omit the proof. When γ is a bounded 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 C r γ.

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

L L L L s L r L I (γs) ≤ I (γr) ≤ I (γ), (resp. I (γ ) ≤ I (γ ) ≤ I (γ), )

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

L r→1− L L r→0+ I (γr) −−−−→ I (γ); I (γr) −−−−→ 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 the Loewner energy measures the “roundness” of a Jordan curve. In particular, the vanishing of the energy of γr as r → 0 expresses the fact that conformal maps asymptotically take small 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 for multichordal SLEs have been obtained by many groups [Car03, Wer04a,BBK05, Dub07, KL07,Law09,MS16a,MS16b,BPW18,PW19], and models the interfaces in 2D statistical mechanics 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 without loss 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 for the curves, where ! 1 2n C = (4.1) n n + 1 n 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. Q 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 where

γj ∈ X (D; xaj , xbj ) for each j ∈ {1, . . . , n}. We endow Xα(D; x1, . . . , x2n) with the relative product topology and recall that X (D; xaj , xbj ) is endowed with the relative topology induced from a Hausdorﬀ metric as in Section 2. Multichordal SLEκ is a random multichord γ = (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 natural definition of multichordal SLE is such that for each j, the law of the random curve γj is the chordal SLEκ in Dˆj, conditioned on the other curves {γi | i 6= j}. Here, Dˆj is the S component of D r i6=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 chain on Xα(D; x1, . . . , x2n) defined by re-sampling the curves from their conditional laws. This idea was already introduced and used earlier in [MS16a, MS16b], where Miller & Sheffield studied interacting SLE curves coupled with the Gaussian free field (GFF) in the framework of “imaginary geometry”.

x2n x1

xb γj j

xaj

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

By Radon-Nikodym derivative: We assume1 that 0 < κ < 8/3. Multichordal SLE can be obtained by weighting n independent SLEκ by c(κ) (3κ − 8)(6 − κ) exp m (γ , . . . , γ ) , where c(κ) := < 0, (4.3) 2 D 1 n 2κ is the central charge associated to SLEκ. The quantity mD(γ) is deﬁned using the Brownian loop loop measure µD introduced by Lawler, Schramm, and Werner [LSW03, LW04]:

n X loop mD(γ) := µD ` ` ∩ γi 6= ∅ for at least p chords γi p=2 Z loop = max #{chords hit by `} − 1, 0 dµD (`) which is positive and finite whenever the family (γi)i=1...n is disjoint. In fact, the Brownian loop measure is an infinite measure on Brownian loops, that is conformally invariant, and 0 loop loop 0 for D ⊂ D, µD0 is simply µD restricted to loops contained in D . When D has non-polar loop boundary, the infinity of total mass of µD comes only from the contribution of small loops. In particular, the summand {` ` ∩ γi 6= ∅ for at least p chords γi} is finite if p ≥ 2 and the chords are disjoint. For n independent chordal SLEs connecting (x1, . . . , x2n), they may intersect each other. However, in this case mD is infinite and the Radon-Nikodym derivative (4.3) vanishes since c < 0. We point out that mD(γ) = 0 if n = 1 (which is not surprising 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 the event that all γj are pairwise disjoint.

20 Remark 4.1. Notice that when κ = 0, c = −∞. The second characterization does not apply and we show its existence and uniqueness by making links to rational functions in the 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 deterministic multichord η = (η1, . . . , ηn) with the property that each ηj is the SLE0 curve in its own ˆ ˆ component (Dj; xaj , xbj ). In other words, each ηj is the hyperbolic geodesic in (Dj; xaj , xbj ), see Remark 1.9. We call a multichord with this property a geodesic multichord. By the conformal invariance of the geodesic property, we assume that D = H without loss of generality. The existence of geodesic multichord for each α follows by characterizing them as minimizers of a lower semicontinuous Loewner energy which is the large deviation rate function of multichordal SLE0+, to be discussed in the next section. Assuming the existence, the uniqueness is a consequence of the following algebraic result.

Theorem 4.2 ([PW20, Thm. 1.2, Prop. 4.1]). If η¯ ∈ Xα(H; x1, . . . , x2n) is a geodesic multichord. The union of η¯, its complex conjugate, and the real line is the real locus of a rational function hη of degree n + 1 with critical points {x1, . . . , x2n}. The rational function ∗ is 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 Cˆ over Cˆ, or equivalently, the ratio of two polynomials. The degree of h is the number of preimages of any regular value. ˆ A point x0 ∈ C is a critical point (equivalently, a branched point) with index k (where k ≥ 2) if

k k+1 h(x) = h(x0) + C(x − x0) + O((x − x0) ) for some constant C 6= 0. In other words, the function is locally a k-to-1 branched cover in the neighborhood of x0. By the Riemann-Hurwitz formula, a rational function of degree n + 1 has 2n critical points if and only if all of its critical points are of index two. The group ! n a b o PSL(2, R) = A = : a, b, c, d ∈ R, ad − bc = 1 c d /A∼−A

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

Proof. The complement Hrη has n+1 components that we call faces. We pick an arbitrary face F and consider a uniformizing conformal map hη from F onto H. Without loss of 0 generality, we consider the former case and assume that F is adjacent to η1. We call F the other face adjacent to η1. Since η1 is a hyperbolic geodesic in Dˆ1, the map hη extends by 0 reﬂection to a conformal map on Dˆ1. In particular, this extension of hη maps F conformally ∗ onto H . By iterating the analytic continuation across all of the chords ηk, we obtain a

21 ˆ meromorphic function hη : H → C. Furthermore, hη also extends to H, and its restriction h | takes values in . Hence, Schwarz reﬂection allows us to extend h to ˆ by setting η R R η C ∗ ∗ ∗ hη(z) := hη(z ) for all z ∈ H .

Now, it follows from the construction that hη is a rational function of degree n + 1, as ∗ −1 exactly n + 1 faces are mapped to H and n + 1 faces to H . Moreover, hη (R ∪ {∞}) is precisely the union of η, its complex conjugate and R ∪ {∞}. Finally, another choice of the face F we started with yields the same function up to post-composition by PSL(2, R) and z 7→ −z. This concludes the proof.

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

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

Assuming the existence of geodesic multichord in Xα(H; x1, . . . , x2n) and observing that two rational functions constructed in Theorem 4.2 are PSL(2, C) equivalent if and only if they are spanhPSL(2, R), z 7→ −zi 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 rational function up to post-composition by a Möbius transformation of Cˆ.

This corollary is a special case of the Shapiro conjecture concerning real solutions to enumerative geometric problems on the Grassmannian, see [Sot00]. In [EG02], Eremenko and Gabrielov ﬁrst proved this conjecture for the Grassmannian of 2-planes, when the conjecture 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 of multichordal SLE0+.

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

n n 1 X 1 X HD(γ) := ID(γj) + mD(γ) − log PD;x ,x , (4.4) 12 4 aj bj j=1 j=1 where I (γ ) = I (γ ) is the chordal Loewner energy of γ (Deﬁnition 2.2) and D j D,xaj ,xbj j j PD;x,y is the Poisson excursion kernel, deﬁned via

0 0 −2 PD;x,y := |ϕ (x)||ϕ (y)|PH;ϕ(x),ϕ(y), and PH;x,y := |y − x| , and where ϕ: D → H is a conformal map such that ϕ(x), ϕ(y) ∈ R.

22 When n = 1, 1 1 H (γ) = I (γ) − log P . D 12 D 4 D;a,b We denote the minimal potential by

α MD(x1, . . . , x2n) := inf HD(γ) ∈ (−∞, ∞), (4.5) γ with inﬁmum taken over all multichords γ ∈ Xα(D; x1, . . . , x2n). One important property of the Loewner potential is that it satisﬁes the following cascade relation 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

H (γ) = H (γ ) + H (γ , . . . , γ , γ . . . , γ ). (4.6) D Dˆj j D 1 j−1 j+1 n

In particular, any minimizer of HD in Xα(D; x1, . . . , x2n) is a geodesic multichord, and HD(γ) < ∞ if and only if {γj} are disjoint chords with ID(γj) < ∞.

Using techniques from quasiconformal mappings and the fact that multichords with ﬁnite potential are comprised with quasichords, we can show (see [PW20, Prop. 3.13]):

Proposition 4.8 ([PW20, Prop. 3.13]). The Loewner potential has the following properties.

α 1. The level set {γ ∈ Xα(D; x1, . . . , x2n) | HD(γ) ≤ c} is compact for any c ≥ MD. 2. There exists a multichord minimizing HD in Xα(D; x1, . . . , x2n).

At this point, we know that the inﬁmum in (4.5) is attained, and there exists a geodesic multichord in Xα(D; x1, . . . , x2n) by Lemma 4.7 and complete the proof of the uniqueness as in Corollary 4.4.

Deﬁnition 4.9. We deﬁne the multichordal Loewner energy

α α ID(γ) := 12(HD(γ) − MD(x1, . . . , x2n))  n   n  X X 0 0 =  ID(γj) + 12mD(γ) − inf  ID(γj) + 12mD(γ ) . γ0 j=1 j=1

This energy is a good rate function for our large deviation result.

κ Theorem 4.10 ([PW20, Thm. 1.4]). The family of laws (Pα)κ>0 of multichordal SLEκ α satisﬁes the large deviation principle in Xα(D; x1, . . . , x2n) with good rate function ID. Remark 4.11. When n = 1, Theorem 4.10 is equivalent to Theorem 2.4.

Remark 4.12. The expression of the rate function can be guessed from the Radon-Nikodym derivative (4.3). We may write heuristically the density of a single SLE as exp(−ID(γ)/κ). Taking the expectation of (4.3) with respect to the distribution of n independent SLE in Q j X (D; xaj , xbj ),    n ind c(κ) 1 X 0 0 E exp mD ∼κ→0+ exp − inf  ID(γj) + 12mD(γ ) 2 κ γ0 j=1

23 since c(κ)/2 ∼ −12/κ. The density of multichordal SLE is thus given by c(κ) Q ID(γ) exp mD(γ) exp − α 2 j κ ID(γ) ∼κ→0+ exp − . ind c(κ) κ E exp 2 mD

4.4 Minimal potential

α To define the energy ID, one could have added to the potential HD an arbitrary constant that depends only on the boundary data (x1, . . . , x2n; α), e.g., one may drop the Poisson kernel terms in HD which also alters the value of the minimal potential. The advantage of introducing the Loewner potential (4.4) is that it allows comparing the potential of geodesic multichords of different boundary data. This becomes interesting when n ≥ 2 as the moduli space of the boundary data is non-trivial. In this section we discuss equations satisfied 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 functions can 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 xbj i and the evolution Vt = gt(xi) of the other marked points satisfy the diﬀerential equations  ∂ W = −12∂ Mα (V 1,...,V aj −1,W ,V aj +1,...,V 2n),W = x ,  t t aj H t t t t t 0 aj 2 (4.7) ∂ V i = ,V i = x , for i 6= a ,  t t i 0 i j Vt − Wt for 0 ≤ t < T , where T is the lifetime of the solution and (gt)t∈[0,T ] is the Loewner ﬂow generated by ηj.

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

n (6−κ)/2κ Y ind c(κ) Zα(H; x1, . . . , x2n) := P ;xa ,xb × Eκ exp mD(γ) , H j j 2 j=1

ind where Eκ denotes the expectation with respect to the law of the independent SLEκ multichord γ. As a consequence of the large deviation principle, we obtain that

κ→0+ α κ log Zα(H; x1, . . . , x2n) −→ −12 MH(x1, . . . , x2n). (4.8)

κ The marginal law of the chord γj in the multichordal SLEκ in Xα(H; x1, . . . , x2n) is given by the stochastic Loewner equation derived from Zα:  √ dW = κdB + κ∂ log Z V 1,...,V aj −1,W ,V aj +1,...,V 2n dt, W = x ,  t t aj α t t t t t 0 aj i 2dt i dVt = i ,V0 = xi, for i 6= aj. Vt − Wt See [PW19, Eq. (4.10)]). Replacing naively κ log Z by −12Mα , we obtain (4.7). α H

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

Proof of Theorem 4.13. Let us illustrate the case when n = 1. For n ≥ 2, we use an induction and a conformal restriction formula which compares the driving function of a single chord under conformal maps. See [PW20, Sec. 4.2] for the proof. When n = 1, the minimal potential has an explicit formula: 1 1 MH(x1, x2) = log |x2 − x1| =⇒ ∂1MH(x1, x2) = . (4.9) 2 2(x1 − x2)

The hyperbolic geodesic in (H; x1, x2) is the semi-circle η with endpoints x1 and x2. We check directly that ∂tWt|t=0 = 6/(x2 − x1). Since hyperbolic geodesic is preserved under 2 its own Loewner ﬂow, i.e., gt(η[t,T ]) is the semi-circle with end points Wt and Vt = gt(x2), we obtain  6 ∂ W = ,W = x ,  t t 2 0 1 Vt − Wt 2  2 2 ∂tVt = 2 ,V0 = x2. Vt − Wt By (4.9), this is exactly Equation (4.7).

Similarly, using the level two null-state Belavin-Polyakov-Zamolodchikov equation satisﬁed by the SLE partition function:  ! κ 2 X 2 (6 − κ)/κ  ∂ + ∂x −  Zα = 0, j = 1,..., 2n, (4.10) 2 xj x − x i (x − x )2 i6=j i j i j the same idea prompts us to ﬁnd the following result, see also [BBK05, AKM20]. Theorem 4.14 ([PW20, Prop. 1.7]). Let U := −12Mα . For j ∈ {1,..., 2n}, we have α H 1 2 6 (∂ U (x , . . . , x ))2 + X ∂ U (x , . . . , x ) = X . (4.11) 2 j α 1 2n x − x i α 1 2n (x − x )2 i6=j i j i6=j i j

The recent work [AKM20] proves further an explicit expression of Uα(x1, ··· , x2n) in terms of the rational function hη associated to the geodesic multichord in Xα(H; x1, . . . , x2n) as considered in Section 4.2. More precisely, following [AKM20], we normalize the rational function 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 2n n Y 2 Y 8 Y Y −4 exp Uα = C (xj − xk) (ζα,l − ζα,m) (xk − ζα,l) , (4.12) 1≤j

Finally let us remark that another reason to include the Poisson kernel to the Loewner potential is that it relates to the more general framework of deﬁning a global notion of Loewner energy in terms of the zeta-regularized determinants of Laplacians. We do not enter into further details here and refer the interested readers to [PW20, Thm. 1.8].

25 5 Large deviations of radial SLE∞

We now turn to the large deviations of SLE∞, namely, when ε := 1/κ using the notation in Deﬁnition 1.2. From (1.6), one can easily show that in the chordal setup, for any ﬁxed t, −1 the conformal map ft = gt : H → H r Kt converges uniformly on compacts to the identity map as κ → ∞ almost surely. In other words, the complement of the SLEκ hull converges for 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 therefore switch 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 Loewner 1 diﬀerential equation driven by a continuous function R+ → S : t 7→ ζt is deﬁned as follows: for all z ∈ D, ζt + gt(z) ∂tgt(z) = gt(z) . (5.1) ζt − gt(z)

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: inf |gs(z) − ζs| > 0}, s∈[0,t] and the growing hulls are given by Kt = {z ∈ D: τ(z) ≤ t}. The solution gt is the conformal 0 t map from Dt := D r Kt onto D satisfying gt(0) = 0 and gt(0) = e . κ Radial SLEκ is the curve γ tracing out the growing family of hulls (Kt)t≥0 driven by a 1 Brownian motion on the unit circle S = {ζ ∈ C : |ζ| = 1} of variance κ, i.e.,

κ iBκt ζt := βt = e , (5.2) where Bt is a standard one dimensional Brownian motion. Radial SLEs exhibit the same phase transitions as in the chordal case as κ varies. In particular, when κ ≥ 8, γκ is almost κ surely space-filling and Kt = γ[0,t]. We now argue heuristically to give an intuition about our results on the κ → ∞ limit and large deviations in [APW20]. During a short time interval [t, t + ∆t] where the flow is well-defined for a given point z ∈ D, we have gs(z) ≈ gt(z) for s ∈ [t, t + ∆t]. Hence, writing −1 the time-dependent vector field (z(ζt + z)(ζt − z) )t≥0 generating the Loewner chain as R −1 κ ( 1 z(ζ + z)(ζ − z) δ κ (dζ)) , where δ κ is the Dirac measure at β , we obtain that S βt t≥0 βt t ∆gt(z) is approximately

Z t+∆t Z Z ζ + gt(z) ζ + gt(z) κ κ κ gt(z) δβs (dζ)ds = gt(z) d(`t+∆t(ζ) − `t (ζ)), (5.3) t S1 ζ − gt(z) S1 ζ − gt(z)

κ 1 κ where `t is the occupation measure (or local time) on S of β up to time t. As κ → ∞, the occupation measure of βκ during [t, t + ∆t] converges to the uniform measure on S1 of total 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 as driving measure, i.e.,

1 Z ζ + gt(z) ∂tgt(z) = gt(z) |dζ| = gt(z). 2π S1 ζ − gt(z) t This implies gt(z) = e z. Similarly, (5.3) suggests that the large deviations of SLE∞ can κ also be obtained from the large deviations of the process (`t )t≥0.

5.2 Loewner-Kufarev equations

The heuristic outlined above leads naturally to consider the Loewner-Kufarev chain driven by measures that we now deﬁne more precisely. Let M(Ω) (resp. M1(Ω)) be the space of Borel measures (resp. probability measures) on Ω. We deﬁne

1 1 N+ = {ρ ∈ M(S × R+): ρ(S × I) = |I| for all intervals I ⊂ R+}.

From the disintegration theorem (see e.g. [Bil95, Theorem 33.3]), for each measure ρ ∈ N+ 1 there exists a Borel measurable map t 7→ ρt from R+ to M1(S ) 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, (ρet)t≥0 of ρ must satisfy ρt = ρet for a.e. t. We denote by (ρt)t≥0 one such disintegration of ρ ∈ N+. For z ∈ D, consider the Loewner-Kufarev ODE

Z ζ + gt(z) ∂tgt(z) = gt(z) ρt(dζ), g0(z) = z. (5.4) S1 ζ − gt(z) Let τ(z) be the supremum of all t such that the solution is well-deﬁned up to time t with gt(z) ∈ D, and Dt := {z ∈ D : τ(z) > t} is a simply connected open set containing 0. The 0 function gt is the unique conformal map of Dt onto D such that gt(0) = 0 and gt(0) > 0. 0 0 t Moreover, it is straightforward to check that ∂t log gt(0) = |ρt| = 1. Hence, gt(0) = e , −t namely, Dt has conformal radius e seen from 0. We call (gt)t≥0 the Loewner-Kufarev chain driven by ρ ∈ N+. −1 It is also convenient to use its inverse (ft := gt )t≥0, which satisﬁes the Loewner PDE: Z 0 ζ + z ∂tft(z) = −zft(z) ρt(dζ), f0(z) = z. (5.5) S1 ζ − z

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 = D r Dt)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 such −t that Dt ⊂ D has conformal radius e and for all 0 ≤ s ≤ t, Dt ⊂ Ds. We now restrict the Loewner-Kufarev chains to the time interval [0, 1] for the topology and simplicity 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 1 1 N[0,1] = {ρ ∈ M1(S × [0, 1]) : ρ(S × I) = |I| for all intervals I ⊂ [0, 1]},

27 endowed with the Prokhorov topology (the topology of weak convergence) and the corresponding 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 uniform convergence of f on compact subsets of D × [0, 1]. (Or equivalently, viewing L[0,1] as the set of processes (Dt)t∈[0,1], this is the topology of uniform Carathéodory convergence.) The following result allows us to study the limit and large deviations with respect to the topology of 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 δ κ (dζ) dt ∈ N converges almost surely to the βt [0,1] uniform 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 the −t radial SLEκ converges almost surely to (e D)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 principle of radial SLE as κ → ∞ boils down to the large deviation principle of δ κ (dζ) dt ∈ N κ βt [0,1] for the Prokhorov topology. For this, we approximate δ κ (dζ) dt by βt

2n−1 κ X κ ρn := µn,i(dζ)1t∈[i/2n,(i+1)/2n) dt, i=0

κ 1 n n where µ ∈ M (S ) is the average of the measure δ κ on the time interval [i/2 , (i+1)/2 ): n,i 1 βt

κ n κ κ µn,i = 2 (`(i+1)/2n − `i/2n ).

κ κ −1 κ We start with the large deviation principle for µn,i as κ → ∞. Let `t := t `t be the average occupation measure of βκ up to time t. From the Markov property of Brownian κ κ κ motion, we have µn,i = `2−n in distribution up to a rotation (by βi/2n ). The following result is a special case of a theorem of Donsker and Varadhan. Deﬁne the functional IDV : M(S1) → [0, ∞] by 1 Z IDV (µ) = |v0(ζ)|2 |dζ|, (5.6) 2 S1 if µ = v2(ζ)|dζ| for some function v ∈ W 1,2(S1) and ∞ otherwise.

DV DV DV Remark 5.4. Note that I is rotation-invariant and I (cµi) = cI (µi) for c > 0. κ Theorem 5.5 ([DV75, Thm. 3]). Fix t > 0. The average occupation measure {`t }κ>0 admits a large deviation principle as κ → ∞ with good rate function tIDV . Moreover, IDV is convex.

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

1 κ DV lim log P[`t ∈ O] ≥ − inf tI (µ); κ→∞ κ µ∈O 1 κ DV lim log P[`t ∈ F ] ≤ − inf tI (µ). κ→∞ κ µ∈F

n κ κ Theorem 5.5 implies that the 2 -tuple (µn,0, . . . .µn,2n−1) satisﬁes the large deviation principle with rate function

2n−1 DV −n X DV In (µ0, . . . , µ2n−1) := 2 I (µi). i=0 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]) Z 1 DV S[0,1]((Dt)t∈[0,1]) := S[0,1](ρ) := I (ρt) dt 0 DV where ρ is the driving measure generating (Dt) and I .

Theorem 5.7 ([APW20, Thm. 1.2]). The measure δ κ (dζ) dt ∈ N satisﬁes the large βt [0,1] deviation 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, then ρt is absolutely continuous with respect to the Lebesgue measure for a.e. t with density 1,2 1 being the square of a function in W (S ). In particular, ρt is much more regular than a Dirac measure. We see once more the regularizing phenomenon from the large deviation consideration.

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] satisﬁes the 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 the outer boundary of SLE16/κ for κ < 4. It suggests that the rate functions of SLE0+ (Loewner energy) and SLE∞ (Loewner-Kufarev energy) are also dual to each other. Let us ﬁrst remark 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 example supports the guess that some form of energy duality holds. Viklund and I [VW20b] investigated the duality between these two energies, and more generally, the interplay with the Dirichlet energy of a so-called winding function. We now describe 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 also deterministic.

29 6.1 Whole-plane Loewner evolution

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

1 1 N := {ρ ∈ M(S × R): ρ(S × I) = |I| for all intervals I}.

The whole-plane Loewner chain driven by ρ ∈ N , or equivalently by its measurable family 1 of disintegration measures R → M1(S ): 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. 0 −t −t (ii) For all t ∈ R, ft(0) = 0 and ft(0) = e (i.e., Dt has conformal radius e ). (s) −1 (s) (iii) For all s ∈ R, (ft := fs ◦ ft : D → Dt )t≥s is the Loewner chain driven by (ρt)t≥s, (s) which satisﬁes (5.5) with the initial condition fs (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.

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

Note that the condition (iii) is equivalent to for all t ∈ R and z ∈ D, Z 0 ζ + z ∂tft(z) = −zft(z) ρt(dζ). (6.1) S1 ζ − z

−t 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 /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 C r {0} if

1. For all t ∈ R, γt is a chord-arc Jordan curve (see Remark 2.17). 1 2. It is possible to parametrize each curve γt by S so that the mapping t 7→ γt is continuous in the supremum norm. 3. For all z ∈ C r {0}, τ(z) < ∞.

Each γt of a foliation is called a leaf.

Deﬁnition 6.2. We deﬁne similarly the Loewner-Kufarev energy on N by Z ∞ DV S(ρ) := I (ρt) dt, −∞ where IDV is given by (5.6).

30 ϕ(z)

0 z

Figure 3: Illustration of the winding function ϕ.

6.2 Energy duality

The following result gives a qualitative relation between ﬁnite Loewner-Kufarev energy measures 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 on 1 S × R with ﬁnite Loewner-Kufarev energy. Then ρ generates a foliation (γt = ∂Dt)t∈R of C r {0} in which all leaves are Weil-Petersson quasicircles.

Every ρ with S(ρ) < ∞ thus generates a foliation of Weil-Petersson quasicircles that continuously sweep out the Riemann sphere, starting from ∞ and moving towards 0, as t ranges from −∞ to ∞. This class of measures seems to be a rare case for which a complete description of the geometry of the generated non-smooth interfaces is possible. We also show a quantitative relation among energies by introducing a real-valued winding −1 function ϕ associated to a foliation (γt)t∈R as follows. Let gt = ft : Dt → D. Since γt is by assumption chord-arc, thus rectifiable, γt has a tangent arclength-a.e. Given z at which γt has a tangent, we define g0(w) ϕ(z) = lim arg w t (6.2) w→z gt(w) where the limit is taken inside Dt and approaching z non-tangentially. We choose the continuous branch of arg which vanishes at 0. Monotonicity of (Dt)t∈R implies that there is no 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 of γt 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 consists of concentric circles centered at 0 whose winding function is identically 0. The following is our main theorem.

Theorem 6.5 (Energy duality [VW20b, Thm. 1.2]). Assume that ρ ∈ N generates a foliation and let ϕ be the associated winding function on C. Then DC(ϕ) < ∞ if and only if S(ρ) < ∞ and

DC(ϕ) = 16 S(ρ).

31 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 trees theorem 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 case of chord-arc foliation, a function defined arclength-a.e. on all leaves need not be defined Lebesgue-a.e., see, e.g., [Mil97]. Thus to consider the Dirichlet energy of ϕ, we consider the 1,2 following extension to Wloc . A function ϕ defined arclength-a.e. on all leaves of a foliation 1,2 (γt = ∂Dt) is said to have an extension φ in Wloc if for all t ∈ R, the Jonsson-Wallin trace

φ|γt of φ on γt deﬁned by the following limit of averages on balls B(z, r) = {w : |w − z| < r} Z φ|γt (z) := lim φ dA (6.3) r→0+ B(z,r) coincides with ϕ arclength-a.e on γt. We also show that if such extension exists then it is unique. The Dirichlet energy of ϕ in the statement of Theorem 6.5 is understood as the Dirichlet energy of this extension.

Theorem 6.5 has several applications which show that the foliation of Weil-Petersson quasicircles generated by ρ with ﬁnite Loewner-Kufarev energy exhibits several remarkable features and symmetries. The ﬁrst is the reversibility of the Loewner-Kufarev energy. Consider ρ ∈ N and the ˆ corresponding family of domains (Dt)t∈R. Applying z 7→ 1/z to C r Dt, we obtain an ˜ evolution family of domains (Dt)t∈R upon time-reversal and reparametrization, which may be described by the Loewner equation with an associated driving measure ρ˜. While there is no known simple description of ρ˜ in terms of ρ, energy duality implies remarkably that the Loewner-Kufarev energy is invariant under this transformation.

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

Proof sketch. The map z 7→ 1/z is conformal and the family of circles centered at 0 is preserved under this inversion. We obtain from the geometric interpretation of ϕ (Remark 6.4) that the winding function ϕ˜ associated to the foliation (∂D˜t) satisﬁes ϕ˜(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(ϕ˜) = DC(ϕ) and conclude with Theorem 6.5.

Remark 6.9. It is not known whether whole-plane SLEκ for κ > 8 is reversible. (For κ ≤ 8, reversibility was established in [Zha15,MS17].) Therefore Theorem 6.8 cannot be predicted from the SLE point of view by considering the κ → ∞ large deviations as we did for the reversibility of chordal Loewner energy in Theorem 2.7. This result on the other hand 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 a necessary condition to be a leaf in the foliation generated by a ﬁnite energy measure. The next 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. We obtain 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 C r γ ﬁxing 0 (resp. ﬁxing ∞). Theorem 6.10 (Characterization of Weil-Petersson quasicircles [VW20b, Thm. 1.4]). The curve γ is a Weil-Petersson quasicircle if and only if γ can be realized as a leaf in the foliation generated by a measure ρ with S(ρ) < ∞. Moreover, we have

IL(γ) = 16 inf S(ρ) + 2 log |f 0(0)/h0(∞)|, ρ where the inﬁmum, which is attained, is taken over all ρ ∈ N such that γ is a leaf of the generated foliation. Remark 6.11. The inﬁmum is only realized for the measure generating the family of equipotentials on both sides of γ, i.e., the image of the circles centered at 0 under f and h. In this case, the winding function is harmonic in C r γ. Note also that this minimum is zero if and only if γ is a circle centered at 0, whereas IL(γ) is zero for all circles. This explains the presence of the derivative terms.

A consequence of generalized Grunsky inequality, see, e.g., [TT06, p. 70-71], shows that log |f 0(0)/h0(∞)| ≥ 0. We obtain: Corollary 6.12 ([VW20b, Cor. 8.7]). Any leaf γ of the foliation generated by ρ satisﬁes IL(γ) ≤ 16 S(ρ).

Another consequence of Theorem 6.10 is the following complex identity for bounded curve analogous to Corollary 3.6 which simultaneously expresses the interplay between Dirichlet energies under “welding” and “ﬂow-line” operations, see Theorem 3.1, 3.4. 1,2 More precisely, we say that a Weil-Petersson quasicircle γ is compatible with φ ∈ Wloc , if the winding function along γ, as deﬁned in (6.2), coincides with the trace φ|γ arclength-a.e. Theorem 6.13 (Complex identity [VW20b, Prop. 1.5]). Let ψ be a complex valued function on C with DC(ψ) = DC(Re ψ)+DC(Im ψ) < ∞ and γ a Weil-Petersson quasicircle separating 0 from ∞ compatible with Im ψ. Let f 0(z)z h0(z)z ζ(z) := ψ ◦ f(z) + log and ξ(z) := ψ ◦ h(z) + log . (6.4) f(z) h(z)

Then we have DC(ψ) = DD(ζ) + DD∗ (ξ). Remark 6.14. We do not use complex conjugate in (6.4) as opposed to Corollary 3.6. The reason is that the deﬁnitions for compatibility and being a ﬂow-line in both theorems have an opposite sign.

7 Summary

Let us end with a table summarizing the results presented in this survey, highlighting the close analogy between concepts and theorems from random conformal geometry and

33 the finite energy/large deviation world. Although some results involving the finite energy objects are interesting on their own from the analysis perspective, we choose to omit from the 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 SLE are great sources generating exciting results in the deterministic world. Vice versa, as finite energy objects are more regular and easier to handle, exploring their properties would also provide a way to speculate new conjectures in random conformal geometry. Rather than an end, we hope it to be the starting point of new development along those lines and this survey can serve as a guide.

SLE/GFF with κ 1 Finite energy Section 1, 2

Chordal SLEκ in (D, x, y) A chord γ with ID,x,y(γ) < ∞ (Deﬁnition 2.2, Theorem 2.4)

Chordal SLE0 in (D, x, y) Hyperbolic geodesic in (D, x, y) (Remark 1.9)

Chordal SLEκ is reversible Chordal Loewner energy is reversible (Theorem 2.7) Jordan curve γ with IL(γ) < ∞

SLEκ loop i.e., a Weil-Petersson quasicircle γ (Theorem 2.14) Section 3 √ Free boundary GFF κΦ on (on ) 2u ∈ E( ), i.e., D (u) < ∞ (2ϕ ∈ E( )) H √ C H H C √ κΦ 2ϕ κ-LQG on quantum plane ≈ e dA(z) e dA(z), ϕ ∈ E(C) √ 2u κ-LQG on quantum half-plane on H e dA(z), u ∈ E(H) √ κ-LQG boundary measure on eu dx, u ∈ H1/2( ) √ R R ≈ e κΦ/2dx

SLEκ cuts an independent A Weil-Petersson quasicircle γ cuts √ quantum plane e κΦdA(z) into ϕ ∈ E( ) into u ∈ E( ), v ∈ E( ∗) and √ √ C H H κΦ1 κΦ2 L ind. quantum half-planes e , e I (γ) + DC(ϕ) = DH(u) + DH∗ (v) (Theorem 3.1) √ Isometric welding of independent κ-LQG Isometric welding of eu dx and ev dx, 1/2 boundary measures on R produces SLEκ u, v ∈ H (R) produces a Weil-Petersson quasicircle (Theorem 3.2) √ Bi-infinite flow-line of eiΦ/χ ≈ ei κΦ/2 is Bi-infinite flow-line of eiϕ is a Weil-

an SLEκ loop Petersson quasicircle (Theorem 3.4) Section 4 α Multichordal SLEκ in D of link pattern α Multichord γ with ID(γ) < ∞ (Deﬁnition 4.9, Theorem 4.10) Multichordal SLE0 in H Real locus of a real rational function i.e., geodesic multichord (Theorem 4.2)

34 α Multichordal SLEκ pure partition function Minimal potential MD Zα of link pattern α (Deﬁnition 4.6, Equation (4.8)) Loewner evolution of a chord in Loewner evolution of a chord in geodesic

multichordal SLEκ multichord (Theorem 4.13) Level 2 null-state BPZ equation Semiclassical null-state equation satisﬁed by Z satisﬁed by Mα (Theorem 4.14) α H Section 5, 6

Loewner chain (Dt)t∈R driven by ρ ∈ N Whole-plane radial SLE16/κ with Loewner-Kufarev energy S(ρ) < ∞. (Corollary 5.9, Deﬁnition 6.2)

Radial mating of trees (unknown) Energy duality DC(ϕ) = 16 S(ρ) (Theorem 6.5)

∂Dt is a Weil-Petersson quasicircle The outer boundary of SLE16/κ for all t ∈ R and L 0 0 is locally a SLEκ curve I (γ) = 16 infρ S(ρ) + 2 log |f (0)/h (∞)| (Theorem 6.10)

Reversibility of SLE16/κ (unknown) Loewner-Kufarev energy is reversible (Theorem 6.8)

References

[AKM20] Tom Alberts, Nam-Gyu Kang, and Nikolai Makarov. Pole dynamics and an integral of motion for multiple SLE(0). arXiv:2011.05714, 2020.

[APW20] Morris Ang, Minjae Park, and Yilin Wang. Large deviations of radial SLE∞. arXiv preprint: 2002.02654, 2020. [BB03] Michel Bauer and Denis Bernard. Conformal ﬁeld theories of stochastic Loewner evolutions. Comm. Math. Phys., 239(3):493–521, 2003. [BBK05] Michel Bauer, Denis Bernard, and Kalle Kytölä. Multiple Schramm-Loewner evolutions and statistical mechanics martingales. J. Stat. Phys., 120(5-6):1125– 1163, 2005.

[BD16] Stéphane Benoist and Julien Dubédat. An SLE2 loop measure. Ann. Inst. Henri Poincaré Probab. Stat., 52(3):1406–1436, 2016. [Bef08] Vincent Beﬀara. The dimension of the SLE curves. Ann. Probab., 36(4):1421– 1452, 2008. [Bil95] Patrick Billingsley. Probability and measure. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, third edition, 1995. A Wiley-Interscience Publication.

35 [Bis90] Christopher J. Bishop. Conformal welding of rectifiable curves. Math. Scand., 67(1):61–72, 1990. [Bis19] Christopher J Bishop. Weil-petersson curves, conformal energies, β-numbers, and minimal surfaces. preprint, 2019. [BPW18] Vincent Beffara, Eveliina Peltola, and Hao Wu. On the uniqueness of global multiple sles. arXiv preprint: 1801.07699, 2018. [Car03] John L. Cardy. Stochastic Loewner evolution and Dyson’s circular ensembles. J. Phys. A, 36(24):L379–L386, 2003. [Dav82] Guy David. Courbes corde-arc et espaces de Hardy généralisés. Ann. Inst. Fourier (Grenoble), 32(3):xi, 227–239, 1982. [Din93] I. H. Dinwoodie. Identifying a large deviation rate function. Ann. Probab., 21(1):216–231, 1993. [DMS14] Bertrand Duplantier, Jason Miller, and Scott Sheffield. Liouville quantum gravity as a mating of trees. arXiv preprint arXiv:1409.7055, 2014. [DS89] Jean-Dominique Deuschel and Daniel W. Stroock. Large deviations, volume 137 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1989. [Dub07] Julien Dubédat. Commutation relations for Schramm-Loewner evolutions. Comm. Pure Appl. Math., 60(12):1792–1847, 2007. [Dub09a] Julien Dubédat. Duality of Schramm-Loewner evolutions. Ann. Sci. Éc. Norm. Supér. (4), 42(5):697–724, 2009. [Dub09b] Julien Dubédat. SLE and the free field: partition functions and couplings. J. Amer. Math. Soc., 22(4):995–1054, 2009. [Dub15] Julien Dubédat. SLE and Virasoro representations: localization. Comm. Math. Phys., 336(2):695–760, 2015. [DV75] Monroe D Donsker and SR Srinivasa Varadhan. Asymptotic evaluation of certain markov process expectations for large time, i. Communications on Pure and Applied Mathematics, 28(1):1–47, 1975. [DZ10] Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. [EG02] Alexandre Eremenko and Andrei Gabrielov. Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry. Ann. of Math. (2), 155(1):105–129, 2002. [EG11] Alexandre Eremenko and Andrei Gabrielov. An elementary proof of the B. and M. Shapiro conjecture for rational functions. In Notions of positivity and the geometry of polynomials, Trends Math., pages 167–178. Birkhäuser/Springer Basel AG, Basel, 2011. [FK04] Roland Friedrich and Jussi Kalkkinen. On conformal field theory and stochastic Loewner evolution. Nuclear Phys. B, 687(3):279–302, 2004. [FW03] Roland Friedrich and Wendelin Werner. Conformal restriction, highest-weight representations and SLE. Comm. Math. Phys., 243(1):105–122, 2003.

36 [Gar07] John B. Garnett. Bounded analytic functions, volume 236 of Graduate Texts in Mathematics. Springer, New York, first edition, 2007. [GHS19] Ewain Gwynne, Nina Holden, and Xin Sun. Mating of trees for random planar maps and Liouville quantum gravity: a survey. 2019. [Gol91] Lisa R. Goldberg. Catalan numbers and branched coverings by the Riemann sphere. Adv. Math., 85(2):129–144, 1991. [JVST12] Fredrik Johansson Viklund, Alan Sola, and Amanda Turner. Scaling limits of anisotropic hastings–levitov clusters. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 48(1):235–257, 2012. [KL07] Michael J. Kozdron and Gregory F. Lawler. The configurational measure on mutually avoiding SLE paths. In Universality and renormalization, volume 50 of Fields Inst. Commun., pages 199–224. Amer. Math. Soc., Providence, RI, 2007. [Law05] Gregory F. Lawler. Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. [Law09] Gregory F. Lawler. Partition functions, loop measure, and versions of SLE. J. Stat. Phys., 134(5-6):813–837, 2009. [Lin05] Joan R. Lind. A sharp condition for the Loewner equation to generate slits. Ann. Acad. Sci. Fenn. Math., 30(1):143–158, 2005. [Loe23] Karl Loewner. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann., 89(1-2):103–121, 1923. [LSW03] Gregory Lawler, Oded Schramm, and Wendelin Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc., 16(4):917–955, 2003. [LSW04] Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32(1B):939–995, 2004. [LW04] Gregory F. Lawler and Wendelin Werner. The Brownian loop soup. Probab. Theory Related Fields, 128(4):565–588, 2004. [Mil97] John Milnor. Fubini foiled: Katok’s paradoxical example in measure theory. The Mathematical Intelligencer, 19(2):30–32, 1997. [MR05] Donald E. Marshall and Steffen Rohde. The Loewner differential equation and slit mappings. J. Amer. Math. Soc., 18(4):763–778, 2005. [MS16a] Jason Miller and Scott Sheffield. Imaginary geometry I: interacting SLEs. Probab. Theory Related Fields, 164(3-4):553–705, 2016. [MS16b] Jason Miller and Scott Sheffield. Imaginary geometry II: reversibility of SLEκ(ρ1; ρ2) for κ ∈ (0, 4). Ann. Probab., 44(3):1647–1722, 2016.

[MS16c] Jason Miller and Scott Sheffield. Imaginary geometry III: reversibility of SLEκ for κ ∈ (4, 8). Ann. of Math. (2), 184(2):455–486, 2016. [MS16d] Jason Miller and Scott Sheffield. Quantum Loewner evolution. Duke Mathemat- ical Journal, 165(17):3241–3378, 2016. [MS17] Jason Miller and Scott Sheffield. Imaginary geometry IV: interior rays, whole-

37 plane reversibility, and space-filling trees. Probab. Theory Related Fields, 169(3-4):729–869, 2017. [Pel19] Eveliina Peltola. Toward a conformal field theory for Schramm-Loewner evolutions. J. Math. Phys., 60(10):103305, 39, 2019. [Pom65] Christian Pommerenke. Über die subordination analytischer funktionen. Journal für die reine und angewandte Mathematik, 218:159–173, 1965. [Pom75] Christian Pommerenke. Univalent functions. Vandenhoeck & Ruprecht, Göt- tingen, 1975. With a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV. [Pom78] Ch. Pommerenke. On univalent functions, Bloch functions and VMOA. Math. Ann., 236(3):199–208, 1978. [PW19] Eveliina Peltola and Hao Wu. Global and local multiple SLEs for κ ≤ 4 and connection probabilities for level lines of GFF. Comm. Math. Phys., 366(2):469–536, 2019.

[PW20] Eveliina Peltola and Yilin Wang. Large deviations of multichordal SLE0+, real rational functions, and zeta-regularized determinants of Laplacians. arXiv preprint arXiv:2006.08574, 2020. [RR94] Marvin Rosenblum and James Rovnyak. Topics in Hardy classes and univalent functions. Birkhäuser, Basel, 1994. [RS05] Steffen Rohde and Oded Schramm. Basic properties of SLE. Ann. of Math. (2), 161(2):883–924, 2005. [RW19] Steffen Rohde and Yilin Wang. The Loewner energy of loops and regularity of driving functions. International Mathematics Research Notices, published online 2019. [Sch00] Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221–288, 2000. [Sch07] Oded Schramm. Conformally invariant scaling limits: an overview and a collection of problems. In International Congress of Mathematicians. Vol. I, pages 513–543. Eur. Math. Soc., Zürich, 2007. [Sem86] Stephen Semmes. A counterexample in conformal welding concerning chord-arc curves. Ark. Mat., 24(1):141–158, 1986. [She16] Scott Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab., 44(5):3474–3545, 2016. [Smi06] Stanislav Smirnov. Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians, volume 2, pages 1421–1451. Eur. Math. Soc., Zürich, 2006. [Smi10] Stanislav Smirnov. Conformal invariance in random cluster models. I. Holo- morphic fermions in the Ising model. Ann. of Math. (2), 172(2):1435–1467, 2010. [Sot00] Frank Sottile. Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro. Experiment. Math., 9(2):161–182, 2000.

38 [SS09] Oded Schramm and Scott Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math., 202(1):21–137, 2009. [ST20] Yuliang Shen and Shuan Tang. Weil-Petersson Teichmüller space II: Smoothness 3 of flow curves of H 2 -vector fields. Adv. Math., 359:106891, 25, 2020. [STW18] Yuliang Shen, Shuan Tang, and Li Wu. Weil-Petersson and little Teichmüller spaces on the real line. Ann. Acad. Sci. Fenn. Math., 43(2):935–943, 2018. [TT06] Leon A. Takhtajan and Lee-Peng Teo. Weil-Petersson metric on the universal Teichmüller space. Mem. Amer. Math. Soc., 183(861):viii+119, 2006. [Var66] S. R. S. Varadhan. Asymptotic probabilities and differential equations. Comm. Pure Appl. Math., 19:261–286, 1966. [VW20a] Fredrik Viklund and Yilin Wang. Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines. Geom. Funct. Anal., 30(1):289– 321, 2020. [VW20b] Fredrik Viklund and Yilin Wang. The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles. arXiv:2012.05771, 2020. [Wan19a] Yilin Wang. The energy of a deterministic Loewner chain: reversibility and interpretation via SLE0+. J. Eur. Math. Soc. (JEMS), 21(7):1915–1941, 2019. [Wan19b] Yilin Wang. Equivalent descriptions of the Loewner energy. Invent. Math., 218(2):573–621, 2019. [Wer04a] Wendelin Werner. Girsanov’s transformation for SLE(κ, ρ) processes, intersection exponents and hiding exponents. Ann. Fac. Sci. Toulouse Math. (6), 13(1):121– 147, 2004. [Wer04b] Wendelin Werner. Random planar curves and Schramm-Loewner evolutions. In Lectures on probability theory and statistics, volume 1840 of Lecture Notes in Math., pages 107–195. Springer, Berlin, 2004. [Wer08] Wendelin Werner. The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc., 21(1):137–169, 2008. [Zha08a] Dapeng Zhan. Duality of chordal SLE. Invent. Math., 174(2):309–353, 2008. [Zha08b] Dapeng Zhan. Reversibility of chordal SLE. Ann. Probab., 36(4):1472–1494, 2008. [Zha15] Dapeng Zhan. Reversibility of whole-plane SLE. Probab. Theory Related Fields, 161(3-4):561–618, 2015. [Zha20] Dapeng Zhan. SLE loop measures. Probability Theory and Related Fields, pages 1–62, 2020.