We construct a conformally invariant random family of closed curves in the
plane by welding of random homeomorphisms of the unit circle given in terms of
the exponential of Gaussian Free Field. We conjecture that our curves are
locally related to SLE$(\kappa)$ for $\kappa<4$.Comment: 5 page

Astala, K.

Jones, P.

Kupiainen, A.

Saksman, E.

English

arXiv

Kari Astala

Peter Jones

Antti Kupiainen

Eero Saksman

CiteSeerX

E.: Random curves by conformal welding

Astala, Kari

Jones, Peter

Kupiainen, Antti

Saksman, Eero

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 257–262Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comComplex Analysis/Mathematical PhysicsRandom curves by conformal welding ✩Courbes aléatoires par soudure conformeKari Astala a,1,4, Peter Jones b,2, Antti Kupiainen a,1,3, Eero Saksman a,1a University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, 00014 University of Helsinki, Finlandb Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven, CT 06510, USAa r t i c l e i n f o a b s t r a c tArticle history:Received and accepted 11 December 2009Available online 24 February 2010Presented by Paul MalliavinWe construct a conformally invariant random family of closed curves in the plane bywelding of random homeomorphisms of the unit circle given in terms of the exponential ofGaussian Free Field. We conjecture that our curves are locally related to SLE(κ) for κ < 4.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éOn construit une famille aléatoire conformément invariante de courbes fermées dans leplan par soudure d’un cercle unité donné en terme d’exponentielle d’un champ libregaussien. On conjecture que nos courbes sont localement reliées à SLE(κ) pour κ < 4.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeDans cette Note, on résume une construction d’une famille de courbes aléatoires. Pour la construction on résout leproblème de soudure pour un ensemble aléatoire d’homéomorphismes hω : T → T, localement invariant par changementd’échelle, obtenant ainsi un ensemble aléatoire de courbes de Jordan. Pour définir h, on identifie le cercle à T = R/Z = [0,1)et on écrit h(t) = τ ([0, t))/τ ([0,1)) où τ = h′ est une mesure de Borel positive τ sans atomes. Notre choix de τ estτ (dt) = limε→0 eβ Xε(t)/Eeβ Xε(t) dt,où β  0 et Xε est une régularisation du Champ Gaussien Libre X sur le cercle i.e. le champ aléatoire avec covarianceEX(t)X(t′) = − log∣∣e2π it − e2π it′ ∣∣.Notre résultat principal est alors :✩ We thank M. Bauer, D. Bernard, S. Rohde and S. Smirnov for discussions and L. Dubois for help in French.E-mail addresses: kari.astala@helsinki.fi (K. Astala), jones@math.yale.edu (P. Jones), antti.kupiainen@helsinki.fi (A. Kupiainen), eero.saksman@helsinki.fi(E. Saksman).1 Supported by the Academy of Finland.2 Supported by NSF.3 Supported by ERC.4 Supported by the EU-network CODY.1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.12.014258 K. Astala et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 257–262Théorème 0.1. Pour β2 < 2 et presque sûrement en ω, la formule (4) définit un homéomorphisme du cercle höldérien, tel que leproblème de soudure possède une solution γ , où γ est une courbe de Jordan délimitant un domaine Ω = f+(D) avec une applicationde Riemann hölderienne f+ . Pour un ω donné, la solution est unique à une transformation de Möbius du plan près.La preuve de ce résultat est basée sur des techniques quasi-conformes. On recherche une solution homéomorphe àl’équation de Beltrami.∂ F∂z= χD(z)μ(z) ∂ F∂z, pour p.tout. z ∈ C. (B)où μ est la dilatation complexe de l’extension de Beurling–Ahlfors f de h à l’intérieur du disque unité. On vérifie que lesapplications de soudures sont alors obtenues avec f− := F |C\D et f+ := F ◦ f −1 sur D. Dans ce cadre, on est très loin dessituations couvertes par la théorie classique, où on suppose ‖μ‖∞ < 1. Par conséquent, il reste un problème non-trivial :dans notre situation non-dégénérée, comment prouve-t-on que l’équation auxilliaire (B) possède une solution homéomorphe,unique à une transformation conforme près ?Les principaux ingrédients de notre preuve sont le résultat de Jones et Smirnov [10] pour ce qui est de l’unicité, et unevariante quantitative de la méthode de Lehto pour ce qui est de l’existence et de la continuité höldérienne locale requise denotre solution de (B). Cela nous conduit à estimer des intégrales de Lehto :LK (w, r, R) :=R∫r1∫ 2π0 K (w + ρeiθ )dθdρρ,où K est la distortion :K (z) = 1 + |μ(z)|1 − |μ(z)| .Notre estimation probabiliste principale nous fournit exactement la borne nécessaire pour obtenir les estimations uniformesvoulues pour la continuité Höldérienne de la solution de (B) : Soit w ∈ T et soit β < √2. Alors il existe b > 0 et δ0 > 0 telsque pour δ < δ0 l’intégrale de Lehto satisfait l’estimationP(L(w,2−N ,1)< Nδ) 2−(1+b)N . (E)1. IntroductionA major breakthrough in the study of conformally invariant random curves in the plane occurred when O. Schramm [14]introduced the Schramm–Loewner Evolution (SLE), a stochastic process which describes such curves growing in a fictitioustime so that the curve of interest is obtained as time tends to infinity. In this note we summarize a different construction [3]of random curves which is stationary i.e. the probability measure on curves is directly defined without introducing anauxiliary time. We carry out this construction for closed curves, a case that is not naturally covered by SLE.Our construction is based on the idea of conformal welding which provides a correspondence between Jordan curves onthe extended plane Ĉ and a set of homeomorphisms of the circle T. Given a Jordan curve Γ ⊂ Ĉ, letf+ : D → Ω+ and f− : D∞ → Ω−,be a choice of Riemann mappings of the unit disk D and its complement onto the components of Ĉ \ Γ = Ω+ ∪ Ω− . ByCaratheodory’s theorem f− and f+ both extend continuously to T = ∂D = ∂D∞ , and thusφ = f −1+ ◦ f− (1)is a homeomorphism of T. In the welding problem we are asked to invert this process; given a homeomorphism φ : T → Twe are to find a Jordan curve Γ and conformal mappings f± onto the complementary domains Ω± so that (1) holds. It isclear that the welding problem, when solvable, has natural conformal invariance attached to it; any image of the curve Γunder a Möbius transformation of Ĉ is equally a welding curve. Similarly, if φ : T → T admits a welding, then so do all itscompositions with Möbius transformations of the disk.We solve the welding problem for a random, locally scale invariant set of homeomorphisms hω : T → T, thereby obtain-ing a random set of Jordan curves. To define h, identify the circle as T = R/Z = [0,1). Given a positive Borel measure τwithout atoms we get a homeomorphism h : [0,1) → [0,1) by:h(t) = τ ([0, t))/τ ([0,1)). (2)It was proposed by the second author some years ago that a natural class of homeomorphisms h is obtained by taking τformally proportional to eβ X(t) dt where β  0 and X is the Gaussian Free Field on the circle i.e. the random field X withcovariance:K. Astala et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 257–262 259EX(t)X(t′) = − log∣∣e2π it − e2π it′ ∣∣. (3)For a rigorous definition one introduces a regularization Xε which is a.s. continuous if ε > 0 and shows that almost surelythe weak limit of Borel measures,τ (dz) = limε→0 eβ Xε(z)/Eeβ Xε(z) dz, (4)exists and defines a non-atomic random Borel measure on [0,1].Our main result is then:Theorem 1.1. For β2 < 2 and almost surely in ω, formulas (2), (4) define a Hölder continuous circle homeomorphism, such that thewelding problem has a solution γ , where γ is a Jordan curve bounding a domain Ω = f+(D) with a Hölder continuous Riemannmapping f+ . For a given ω, the solution is unique up to a Möbius map of the plane.The “critical (inverse) temperature” βc =√2 corresponds to loss of continuity of the maps h. For β  βc the limit (4)is zero almost surely. Based on theoretical physics [9] one may conjecture that a corresponding limit of the normalizedmeasures τε/τε([0,1]) is nontrivial also for β  βc and atomic for β > βc thereby giving rise to a discontinuous map h. Thisphase transition is closely connected to the one observed in two dimensional Liouville Quantum Gravity [7] where a twodimensional version of our measure τ is considered.We conjecture that the curves γ locally resemble SLE(2β2) (see also [8] for arguments to this direction). The case β = βc ,presumably corresponding to SLE(4), is not covered by our analysis.It would also be of interest to understand the connection of our weldings to those arising from stochastic flows studiedin the interesting work [1]. In [1] a program was set up for studying weldings that correspond to Hölder continuous randomhomeomorphisms, but the boundary behaviour of the welding maps and hence the existence and uniqueness of the weldingwas left open.2. Beltrami equationA powerful way to solve the welding problem goes by using the Beltrami equation. Assume a homeomorphism φ : T → Tis extended to a locally quasiconformal map f : D → D, i.e. f ∈ C(D) is a homeomorphism with ∇ f locally integrable in Dand satisfying,∂ f∂z= μ(z) ∂ f∂z, for a.e. z ∈ D, (5)with supz∈K |μ(z)| < 1 for K ⊂⊂ D. One then considers the modified equation:∂ F∂z= χD(z)μ(z) ∂ F∂z, for a.e. z ∈ C. (6)Suppose we can find a solution F to (6) which is a homeomorphism of Ĉ. Then Γ = F (T) is a Jordan curve. Moreover,as ∂z F = 0 for |z| > 1, we can set f− := F |D∞ and Ω− := F (D∞) to define a conformal mappingf− : D∞ → Ω−.To get the mapping f+ note that both f and F solve the Beltrami equation (5) in the unit disk D. By the uniquenessproperties of Eq. (5) we conclude thatF (z) = f+ ◦ f (z), z ∈ D, (7)for some conformal mapping f+ : D = f (D) → Ω+ := F (D). Then, on the unit circle,φ(z) = f |T(z) = f −1+ ◦ f−(z), z ∈ T, (8)and we have found a solution to the welding problem.To carry out this set of ideas we observe first that any homeomorphic self map φ of the circle can be extended to alocally quasiconformal map f : D → D via the Beurling–Ahlfors extension. However, a highly nontrivial problem remains:when does the auxiliary equation (6) have a locally quasiconformal solution and when is this unique up to a conformalmap?A classical case where this question can be answered positively is the uniformly elliptic one where there is an extensionwith ‖μ‖∞ < 1. This in turn will be true if φ is quasisymmetric. In our case these conditions do not hold, and we are forcedoutside the uniformly elliptic PDE’s and need to study (6) with strongly degenerate coefficients with only |μ(z)| < 1 almosteverywhere.260 K. Astala et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 257–2623. Existence: Lehto methodWe use the method due to Lehto [12] to show the existence of homeomorphic solutions to (6). This approach is basedon controlling the conformal moduli of images of annular regions. To recall his result, define the distortion function,K (z) = 1 + |μ(z)|1 − |μ(z)| ,corresponding to the complex dilatation μ = μ(z). Given an annulus A(w, r, R) := {z ∈ C: r < |z − w| < R} define the Lehtointegral:L(w, r, R) :=R∫r1∫ 2π0 K (w + ρeiθ )dθdρρ. (9)Lehto’s theorem (see [2, p. 584]) states then that if K (z) is locally integrable, and if for some R0 > 0 the Lehto integralsatisfiesL(z,0, R0) = ∞, for all z ∈ C (10)then the Beltrami equation (6) admits a homeomorphic W 1,1loc -solution F : C → C.We need actually a stronger result on the Lehto integrals to obtain Hölder continuity of the solution. The Lehto integralcontrols the geometric distortion of an annulus under a locally quasiconformal map. Indeed, for a quasiconformal map f :diam(f(B(w, r))) 16 exp(−2π2L(w, r, R)) diam( f (B(w, R))).Eq. (6) is solved by considering a regularized uniformly elliptic equation where μ is replaced by (1 − ε)μ. Since the corre-sponding solutions Fε are conformal in D∞ the diameters diam(Fε(B(w,1))) for w ∈ D are uniformly bounded by Koebe.Thus an estimate,L(w, r,1)  a log 1/r,leads to diam(Fε(B(w, r)))  Cr2π2a i.e. to Hölder continuity of Fε uniformly in ε. Our main probabilistic estimate then is:Theorem 3.1. Let w ∈ T and let β < √2. Then there exists b > 0 and δ0 > 0 such that for δ < δ0 the Lehto integral satisfies theestimateP(L(w,2−N ,1)< Nδ) 2−(1+b)N . (11)This estimate suffices to prove the existence and Hölder continuity of the solution to (6). First, only annuli centered atw ∈ ∂D need to be considered. Second, the b > 0 allows us to cover, for each integer N , ∂D by balls Bi of radii 2−(1+ 12 b)Nsuch that for α > 0diam(F (Bi)) C2−αNfor all i, with probability 1 − O(2− 12 bN). A Borel–Cantelli argument then gives an a.s. Hölder continuity of F .4. Uniqueness of the weldingAn important issue of the welding is its uniqueness, that the curve Γ is unique up to composing with a Möbiustransformation of Ĉ. This would follow from the uniqueness of solutions to the Beltrami equation (6), up to a Möbiustransformation. Unfortunately the condition (10) in Lehto’s theorem is much too weak to imply this. However, in our casethe uniqueness of solutions to the Beltrami equation (6) is equivalent to the conformal removability of the curve F (T).Indeed, suppose that we have two pairs f± and g± of solutions to Eq. (1). Then the formulaΨ (z) ={g+ ◦ ( f+)−1(z) if z ∈ f+(D),g− ◦ ( f−)−1(z) if z ∈ f−(D∞)defines a homeomorphism of Ĉ that is conformal outside Γ = f±(T). Since Γ is a Hölder curve by Theorem 3.1 we caninvoke the result of Jones and Smirnov in [10] that Hölder curves are conformally removable i.e. that Ψ extends conformallyto the entire sphere. Thus it is a Möbius transformation and uniqueness of the welding follows.K. Astala et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 257–262 2615. A large deviation estimateTheorem 3.1 follows from a large deviation estimate for weakly correlated random variables. Let ρ = 2−p where wechoose p large. Let Lk = L(w,ρk,2ρk) so thatL(w,2−Np,1)N∑k=1Lk.For p large Lk are Lehto integrals in well separated annuli (in logarithmic scale). Estimate (11) follows then from theinequalityP(N∑k=1Lk < Nδ)< ρ(1+b)N . (12)The bound (12) is a large deviation estimate and to prove it we establish two facts: that (i) the random variables Lk are(exponentially) weakly correlated and (ii) uniformly in k, P(Lk < ε)  Cε. These facts in turn rely on three ingredients: (a) anextension of φ to f : D → D with good local distortion bounds in terms of the random measure τ , (b) sharp probabilisticbounds for τ and (c) a decomposition of the free field in terms of random fields localized in scale space.For (a) we use the classical Beurling–Ahlfors extension [6]. We pave D by Whitney cubes {C I }I∈D indexed by dyadicintervals I ⊂ ∂D with diam(C I ) and dist(C I , I) comparable to |I|. Then, extending results by Reed on the Beurling–Ahlforsextension [13], for z ∈ C I we have the local distortion bound,K (z)  C∑J , J ′τ ( J )τ ( J ′), (13)where J , J ′ run through dyadic intervals of size 2−4|I| lying in I and its dyadic neighbours. The virtue of this bound is thatthe resulting lower bound for Lehto integral Lk depends mostly on the ratiosτ ( J )τ ( J ′) for J , J′ dyadic intervals of size O(2−kp)and of distance O(2−kp) from w . Thus we need to understand the sizes and mutual correlations of such ratios.For (b) we use results by Bacry and Muzy [4] and Kahane [11] on multiplicative cascades (we refer the reader to [5]for an extensive discussion of random multifractal measures). The most crucial facts are that for β <√2 the measure τ isnon-atomic and for any interval I ,τ (I) ∈ Lp(ω), p ∈ (−∞,2/β2). (14)Hence in particular the ratios in (13) are in L p(ω) for p ∈ [1,2/β2). These facts are used in the proof of statement (ii) above.The fact that we may choose p > 1 is crucial for our analysis and is the source for the restriction to β <√2.Finally for (c), to understand the correlations between the Lk i.e. between the ratiosτ ( J )τ ( J ′) on scale 2−kp we use arepresentation due to Bacry and Muzy [4] of the free field X . It allows to decompose X asX =∞∑k=0ζk, (15)where ζk are mutually independent a.s. continuous fields with ζk(x) independent from ζk(y) for |x − y| > O(2−kp). Thisdecomposition leads to the following lower bound,Ln  mn exp(n−1∑k=02−ap(n−k)tn,k)(1 +∞∑k=n+12−ap(k−n)n,k)−1, (16)for a > 0. The main contribution here is the scale 2−np contribution mn . The positive random variables mn are i.i.d. andsatisfy the condition P(mn < ε)  Cε. Thus their sum∑mn satisfies the estimate (12).The corrections tn,k  0 and n,k  0 represent correlations between scale 2−np and scale 2−kp and are multiplied withweights exponentially small in |n − k|. Further, tn,k has Gaussian tails:P(tn,k > u)  ce−u2/c,and the n,k has a power law tail:P(n,m > λ)  Cλ−q,for q > 1. Moreover, tn,k and tn,′k′ are independent if k = k′ and n,m and n′,m′ are independent if n > m′ or n′ > m. Theseproperties suffice to show that the estimate (12) extends from the mn to the Ln .262 K. Astala et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 257–262References[1] H. Airault, P. Malliavin, A. Thalmaier, Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by acontinuity method along stochastic flows, J. Math. Pures Appl. 83 (2004) 955–1018.[2] K. Astala, T. Iwaniec, G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, vol. 47,Princeton University Press, 2009.[3] K. Astala, P. Jones, A. Kupiainen, E. Saksman, Random conformal weldings, http://arxiv.org/abs/0909.1003, 2009.[4] E. Bacry, J.F. Muzy, Log-infinitely divisible multifractal processes, Comm. Math. Phys. 236 (2003) 449–475.[5] J. Barral, B. Mandelbrot, Random multiplicative multifractal measures. I–III, in: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot,Part 2, in: Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 17–52.[6] A. Beurling, L.V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956) 125–142.[7] B. Duplantier, S. Sheffield, Liouville quantum gravity and KPZ, http://arxiv.org/abs/0808.1560, 2008.[8] B. Duplantier, S. Sheffield, Duality and the Knizhnik–Polyakov–Zamolodchikov relation in Liouville quantum gravity, Phys. Rev. Lett. 102 (150603)(2009), 4 pp.[9] Y.V. Fyodorov, J.P. Bouchaud, Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential, J. Phys. A:Math. Theor. 41 (2008) 372001.[10] P. Jones, S. Smirnov, Removability theorems for Sobolev functions and quasiconformal maps, Arkiv för Matematik 38 (2000) 263–279.[11] J.-P. Kahane, Sur le chaos multiplicatif, Ann. Sci. Math. Québec 9 (1985) 435–444.[12] O. Lehto, Homeomorphisms with a given dilatation, in: Proceedings of the Fifteenth Scandinavian Congress, Oslo, 1968, in: Lecture Notes in Mathe-matics, vol. 118, Springer, 1970, pp. 58–73.[13] T. Reed, On the boundary correspondence of quasiconformal mappings of domains bounded by quasicircles, Pacific J. Math. 28 (1969) 653–661.[14] O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000) 221–288.

Random curves by conformal welding

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Random Curves by Conformal Welding

Random Curves by Conformal Welding

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