RANDOM CONFORMAL WELDING TOWARDS CRITICALITY

Transcription

1 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY NICOLAE TECU Abstract. In this rough draft we outline an extension of the theorem of Astala, Jones, Kupiainen and Saksman to criticality. 1. Introduction We introduce an extension of a theorem Astala, Jones, Kupiainen and Saksman ([1]) to criticality. The authors of [1] constructed a family of random Jordan curves in the plane by solving the conformal welding problem with a random homeomorphism. This random homeomorphism arises from a random measure. The random measure is constructed by a limiting process via the exponentiated Gaussian Free Field (its restriction to the unit circle). The algorithm depends on one parameter, inverse temperature, and provides a random Jordan curve for each inverse temperature less than a certain critical value. In this article we extend the construction to criticality. While Astala, Jones, Kupiainen and Saksman used a white noise representation for the Gaussian Free Field, we use a vaguelet representation. Over the last decades there has been an interest in conformally invariant fractals which could arise as scaling limits of discrete random processes in the plane. We provide a family of conformally invariant closed curves. One of the most important example of conformally invariante fractals is Schramm-Loewner Evolution (introduced by Schramm, [17]). One version describes a random curve evolving in the disk from a point on the boundary to an interior point. The second version describes a curve evolving in the upper half plane from one point on the boundary to another point on the boundary (typically infinity). The construction depends on a parameter κ [0, 8]. It has been shown that the scaling limits of some discrete processes are SLE κ curves. For example, Lawler, Schramm and Werner ([12]) proved that loop erased random walk converges to SLE 2. Another example is the percolation exploration process which to converges to SLE 6 on a triangular lattice as Date: 27 January

2 2 NICOLAE TECU proven by Smirnov ([22]). For more information on SLE κ and an introduction to percolation we refer the reader to G. Lawler s book ([11]) and W. Werner s notes([23]). Unlike SLE κ, the random curves constructed in the present article and in [1]do not evolve in the plane and are closed. While they do not locally look like SLE κ (a result of Binder and Smirnov, reported by Sheffield in [21]), they are closely related as Sheffield has shown in [21]. He welded together two quantum surfaces to get SLE κ for 0 < κ < 4. A quantum surface is essentially a disk with a random measure constructed via a Gaussian Free Field in a similar way to [1]. In our contruction as well as in ([1]) we essentially weld one quantum surface to a deterministic disk. We believe the contruction presented in this paper can be extended to two quantum surfaces. The Gaussian Free Field (random distribution) has been extensively studied in the physics and mathematics literature. For a mathematical introduction see [20], and the introductions of [18],[19]. The welding homeomorphism that we use arises from a random measure whose density (in a generalized, limiting sense) is the exponentiated Gaussian Free Field. Such measures appears also in the work of Sheffield ([21]), Duplantier and Sheffield(e.g. [6]). This measure is a type of multiplicative cascade as introduced by Mandelbrot ([13]) and studied, among others, by Kahane([8]), Kahane and Peyriere ([9]), Bacry and Muzy([3]), Robert and Vargas([16]). Multiplicative cascades appear also in mathematical finance as an important part of the Multifractal Model for Asset Returns introduced by Mandelbrot, Fisher and Calvet ([14]). Acknowledgments I would like to thank my academic advisor, Peter W. Jones, for suggesting this problem and for his support over the course of my doctoral studies. 2. Statement of the theorem Following [1] we can define the restriction of the (random distribution) Gaussian Free Field on the circle by: A n cos(2πnθ) + B n sin(2πnθ) X = (1) n n=1 where A n N(0, 1) B n are independent. In what follows we will use the alternative representation (following an idea of Peter Jones) X = I A I ψ I

3 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 3 where A I N(0, 1) are independent and {ψ I } are periodized halfintegrals of wavelets. They are called vaguelets and appear for example in [5]. Vaguelets satisfy essentially the same properties as wavelets. We can now consider a sequence t n t c (critical inverse temperature) and define the following random measure dν = e I (a Iψ I (θ) t I ψ 2 I (θ)/2) dθ (2) where a I N(0, t I ). The equality should be understood as a limit. Given conditions on the convergence of t n t c, the limiting measure is almost surely a non-zero, finite, non-atomic, singular measure. More precisely we take t n = t c 2 k for all n [n k, n k+1 ), where n k k2 2k. The critical value is t c = 2. The next step is to define the random homeomorphism h : [0, 1) [0, 1): h(θ) = ν([0, θ))/ν([0, 1)) for θ [0, 1). (3) Theorem 1. Almost surely, formula 3 defines a continuous circle homeomorphism, such that the welding problem has a solution γ. The curve γ is a Jordan curve and bounds a domain Ω = f + (Ω) with Riemann mapping f + having the modulus of continuity better than δ(t) = e log 1 t. For a given realization ω, the solution is unique up to Moebius transformations. As mentioned above, this theorem is an extension of the result of Astala, Jones, Kupiainen, Saksman. In their paper, the variances t n were all equal to a value strictly less than the critical one. In addition, the Riemann mapping f + was Hoelder continuous. By contrast, the variances here tend to the critical value, and the Riemann mapping satisfies a weaker modulus of continuity. Astala, Jones, Kupiainen, Saksman prove the theorem by first showing the random measure has the right properties and then solving a degenerate Beltrami equation using the Lehto criterion (which says, roughly, that infinitely many scales with good distortion ensure the existence of a solution). In order to apply this criterion they have to decouple the distortion and prove that, by and large, the different scales behave roughly independently. In this paper we combine the properties of the vaguelets with a martingale square function argument to prove the existence of the random measure µ. We also describe several of its properties. We then do a decoupling, a stopping time argument and a modulus estimate to ensure the distortion in the Beltrami equation does not (with high probability) diverge too fast. Finally, one has to do a careful decoupling of the

4 4 NICOLAE TECU Gaussian Free Field (via the vaguelets) to show that the scales are, by and large, independent. 3. Conformal welding Our goal is to contruct a random Jordan curve in the plane. We accomplish this by solving the conformal welding problem with a random homeomorphism. Let T, D and D be the unit circle, open unit disk and the complement of the closed unit disk respectively. The conformal welding problem is as follows. Let φ be a homeomorphism on S 1. We seek two Riemann mappings f + : D Ω + and f : D Ω onto the complement of a Jordan curve Γ such that φ = (f + ) 1 f. We find these by solving the Beltrami equation. Assume that φ = f T where f W 1,2 loc (D, D) C(D) is a homeomorphism and a solution of the Beltrami differential equation f z = µ(z) f z, z Ω = D. µ is called Beltrami coefficient. If the following Beltrami equation F z = 1 D(z)µ(z) F z, z Ω = D. has a unique (normalized) solution F, we can take f = F D. The uniqueness implies F can be factored as F = f + f for a conformal mapping f + : D F (D). Then we will have φ = (f + ) 1 f. The Beltrami equation admits unique (normalized) solutions whenever µ < 1 (µ is called uniformly elliptic). The mappings φ which can be written as f T with µ uniformly elliptic are called quasisymmetric and satisfy K = sup s,t <. φ(t+s) φ(t) φ(t s) φ(t) In our setting (and the one of Astala, Jones, Kupiaien, Saksman), the mapping φ is not quasisymmetric. While it can be written as the restriction of a mapping f, the corresponding µ is not uniformly elliptic. We will use a version of Lehto s theorem to prove existence of solutions to the Beltrami equation. Define K = 1+ µ and set 1 µ L K (z, r, R) = R r 1 dr 2π K(z + ρe 0 iθ )dθ r Theorem 2 (Lehto, [2], p.584). Suppose µ is measurable, compactly supported and satisfies µ(z) < 1 almost everywhere on C. If the (4)

5 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 5 distortion function K = 1+ µ 1 µ is locally integreable and for some R 0, the Lehto integral satisfies: then the Beltrami equation L K (z, 0, R 0 ) =, z C (5) f z = µ(z) f z, z C. has a homeomorphic solution in W 1,1 loc. We will be using a version of this theorem to prove the main result. Lehto s result says that if one can find around every point an infinite number of annuli whose conformal modulus is distorted only a little bit by a mapping with Beltrami coefficient µ, then the Beltrami equation has a solution. We will prove that we can indeed find an infinite number of good annuli around every point. Astala, Jones, Kupiainen, Saksman worked directly with the Lehto integral above. We will work with moduli of annuli. The uniqueness of the welding will be a consequence of a Jones, Smirnov([7]) /Koskela, Nieminen([10]) conformal removability result. 4. Vaguelets Consider a wavelet basis {Φ j,l } of L 2 (R) with mother wavelet Φ : R R. Following [5] set φ(t) = 1 e itω Φ(ω) ω 1/2 dω 2π The vaguelette φ satisfies the following properties: R φ(t) C 1 (1 + t ) 2, t R, φ(t)dt = 0 φ(t) φ(s) C 2 t s Following Y. Meyer ([15]) consider the periodized functions: Ψ j (θ) = 2 j/2 Φ(2 j (θ k)) The functions {1} {Ψ j (θ k2 j )} j,0 k<2 j form a periodic orthonormal wavelet basis of L 2 (S 1 ).

6 6 NICOLAE TECU For a function on the torus f(θ) f(n)e 2πinθ we have ( )f (2πn) 2 f(n)e 2πinθ so we may define the operator ( ) 1/4 by ( ) 1/4 f(n) = 1 2π n f(n) Define now the periodic vaguelette ψ(θ) by ψ(n) = 1 2π n Ψ(n) where Ψ is the periodic wavelet. Define 1 ψ j,l (n) := Ψ( l2 j ) j (n) (6) 2π n The vaguelets ψ j,l are periodized versions of the φ j,l and have essentially the same properties. While the wavelets are a basis for L 2 (S 1 ), the vaguelets are a basis for H 1/2 (S 1 ) (half a derivative in L 2 ). In particular we have the following: 1 2π n Φ j,l (n) ψ j,l 2 L 2 (S 1 ) = n 0 We also have the following: ψ j,l (θ) dθ C2 j Lemma 3. For the family of vaguelets defined above the following relations hold: there is a constant C 0 such that for all dyadic I, all l and all θ ψj(θ) 2 l ln 2 C 0 J 2 l ψj(θ) 2 C 0 J 2 l,j 3I where 3I is the set formed by the dyadic interval I and its left and right neighbors of the same size. 5. Gaussian Free Field via vaguelets For a good introduction to the GFF see [20]. The GFF we consider is a Gaussian random variable on the infinite dimensional Sobolev space H 1/2 (half a derivative in L 2 ). This is the trace on S 1 of the 2- dimensional GFF that appears in [20], which is a Gaussian random variable on the Sobolev space H 1 = W 1,2 (one derivative in L 2 ).

7 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 7 The trace on S 1 of the 2-dimensional GFF can be defined as the random distribution: A n cos(2πnt) + B n sin(2πnt) X = (7) n n=1 where A n N(0, 1) B n are independent. This representation is obtained by using the Fourier basis for H 1/2 (S 1 ). We can also consider a wavelet basis {φ I } for L 2 (S 1 ). The image under ( ) 1/4 is a vaguelet basis {ψ I } for H 1/2. Then we have that up to a probability preserving transformation this can be rewritten as X = I A I ψ I where A I N(0, 1). We write the GFF as 6. Random measure X = I a I ψ I (θ) (8) where a I N(0, 1) and ψ I = I 1/2 φ I. Now we consider the ( measures ν k = e Sk(t) dt, where S k+1 (t) = S k (t)+ I: I [2 n k+1,2 (n k +1) ] ai ψ I (t) t k+1 2 ψ2 I (t)). Here a I are independent centered Gaussian random variables of variance t k+1 for I [2 n k+1, 2 (n k +1) ]. The sequence {t k } is increasing to t c. Denote F k = dν [0,1] k = [0,1] esk(t) dt. It is easy to see {F k } is an L 1 martingale and hence it has a limit F 0 (almost sure convergence). We want to prove this martingale is uniformly integrable to ensure the F k F 0 in L 1. This and Kolmogorov s zero-one law imply F 0 is almost surely nonzero. Theorem 4. Take t k = t c 2 k and n k (k + 1)2 2(k+1). Then the martingale F k satisfies E[F k log(1 + F k )] < C. We now list the main properties of the random measure ν = lim k ν k. Theorem 5. The limiting measure satisfies: For any subinterval I the random variable ν(i) has all negative moments. Almost surely, for all intervals I, ν(i) > 0. Almost surely, for all intervals I we have ν(i) I a k, where a k corresponds to the t k for which I [2 n k, 2 n k 1). In particular ν(i) e log 1 I.

8 8 NICOLAE TECU In particular, this measure is non-atomic and non-zero on any interval. 7. Random conformal welding We use the random measure constructed above to define a random homeomorphism φ on S 1. Our goal is to solve the conformal welding problem for this homeomorphism. Let h(x) = ν([0,x]), x [0, 1) and extend it periodically to R by ν([0,1]) setting h(x + 1) = h(x) + 1. Extend h to the upper half plane by setting (following Ahlfors-Beurling; see e.g. [2]) 1 F (x+iy) = 1 (h(x+ty)+h(x ty))dt+i (h(x+ty) h(x ty))dt (9) for 0 < y < 1. This function equals h on the real axis and it is a continuously differentiable homeomorphism. For 1 y 2 define F (z) = z + (2 y)c 0, where c 0 = 1 h(t)dt 1/2. For y > 2 define 0 F (z) = z. We also have F (z + k) = F (z) + k. On the unit circle we define the random homeomorphism: and the mapping: φ(e 2πix ) = e 2πih(x). 1 Ψ(z) = exp(2πif (log z/2πi)), z D is the extension of φ to the disk. The distortions of F and Ψ are related by K(z, Ψ) = K(w, F ), z = e 2πiw, w R 2 +. In addition define µ(z) = zψ, z D; µ(z) = 0, z / D. z Ψ Our goal is to solve the Beltrami equation with this Beltrami coefficient. Theorem 6. Almost surely there exists a random homeomorphic W 1,1 loc solution f : C C to the Beltrami equation z f = µ z f, which satisfies f(z) = z + o(1) as z and whose restriction to T has the modulus of continuity ω(t) e log 1 t. Proof. The proof is essentially the same as in [1]. In this proof we use estimate (13) We start by considering for each n an [ ρ 1 n ρn (1+bn/2)Nn ] =: r n net of points on [0, 1] and denote ζ n,k = exp(2πik/r n ) for k {1,..., r n }.

9 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 9 Set also G n = {ζ n,1,..., ζ n,rn }. Any other point on D is at distance at most ρ n ρ (1+bn/2)Nn n from G n. Define the event A n,k := {Mod(F (A(ζ n,k, ρ n ρ (1+bn/2)Nn n, ρ n ))) < c n N n } Now set A n = k A n,k. Since n=1 P (A n ) n r n k=1 P (A n,k ) n r n ρ n ρ (1+bn)Nn n n ρ Nnbn/2 n n Borel-Cantelli tells us that almost every realization ω in in the complement of n>n0 (ω)a n. Consider the approximations µ l = l µ to µ. For each l denote l+1 by f l the normalized (random) solution of the Beltrami equation with coefficient µ l and such that f l (z) = z + o(1) as z. In other words f l is a quasiconformal homeomorphism of C. We want to prove that almost surely the family {f l } is equicontinuous. Outside D all these mappings are conformal and equicontinuity follows from Koebe s theorem. Equicontinuity inside D follows from the fact that at any point inside the disk the distortion is determined by the measure ν on finitely many intervals and hence it is bounded. To prove equicontinuity on S 1 we consider the functions F l (z) = f l (e 2πiz ) For P -a.e. ω we have n Mod(F l (A(ζ n,k, ρ n ρ (1+bn/2)Nn n, 1))) > c i N i, l. Lemma 2.3 in [1] gives: i>i 0 (ω) diam(f (B(ζ, R))) diam(f (B(ζ, r))) 1 exp(πmod(f (A(ζ, r, R)))) 16 for any F quasiconformal. Fix one realization ω. Putting together the last two inequalities (F l is quasiconformal for any l). 1 <. 2Nn/2 diam(f l (B(ζ n,k, ρ n ρ (1+bn/2)Nn n ))) 16diam(F l (B(ζ n,k, 1)))e n i=1 c in i e i 0 (ω) i=1 k 0 (ω)c i N i which gives us the equicontinuity. Arzela-Ascoli now gives us a subsequence of {f l } which converges uniformly on compact sets to a function f : C C. We now show that this sequence can be picked such that f is actually a homeomorphism. To this end consider the inverse functions g l = f 1 l. These functions

10 10 NICOLAE TECU satisfy the estimate: g l (z) g l (w) 16π 2 z 2 + w µ l (ζ) da(ζ) D 1 µ l (ζ) log(e + 1 z w ) Since 1+ µ l(ζ) 1 µ l K(ζ) and K L 1 (D) almost surely we immediately (ζ) have that the sequence {g l } is equicontinuous. The modulus of continuity is given by the relation between ρ n ρ n (1+bn)Nn and e n i=1 c in i. We give now an upper bound for the distortion of µ that we will use. We also introduce necessary notation. Let D n be the collection of dyadic intervals of size 2 n. For a dyadic interval I, let j(i) be the union of I and it s to neighbors of the same size. Set C I = {(x, y) x I, 2 n 1 y 2 n }. Following [1] let J = {J 1, J 2 } and set In addition, define and δ ν (J) = ν(j 1) ν(j 2 ) + ν(j 2) ν(j 1 ) (10) J (I) = {J = (J 1, J 2 ) : J i D n+5, J i j(i)} (11) K ν (I) = J J (I) δ ν (J) (12) The distortion of Ψ is the same as the distortion of F (the points are mapped appropriately). In the upper half of the square with base I(denoted by C I ) the distortion of F is bounded by C 0 K ν (I), for a universal constant C 0. As a consequence, studying the distortion of µ is really about studying the doubling properties of the random measure ν in j(i). In the rest of this paper we will only use K ν. 8. Main probabilistic estimate We want to prove that almost surely we can find infinitely many annuli around each point on the unit circle which are not distorted much by a mapping with Beltrami coefficient µ. We will need the following theorem: Theorem 7. There are sequences ρ n, ρ n, N n, b n, c n such that: N n P ( Mod(G(A(z, ρ n ρ i n, 2 ρ n ρ i n))) < c n N n ) ρ n ρ n (1+bn)Nn (13) i=1

11 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 11 for any z T and any mapping G with Beltrami coefficient µ. We apply this theorem to a net of points on the unit circle and use the Borel-Cantelli theorem to get the desired statement that almost surely around every point on the unit circle there are infinitely many annuli distorted only a little bit. This theorem replaces the Lehto estimate (theorem 4.1) from Astala, Jones, Kupiainen, Saksman([1]). While their estimate covered scales one to ρ n, this estimate deals with the scales in chunks. The theorem is a statement about distortion. Ideally the distortion in one scale would be independent of the distortion in another scale. However, this is not the case, each scale being correlated with every other scale. Fortunately, the correlations decay exponentially. The setting here is more complicated than the one in [1]. They used a representation of the Gaussian Free Field in terms of white noise W - a centered Gaussian process, indexed by sets of finite hyperbolic area measure in the upper half-plane and with covariance structure given by the hypebolic area measure of the intersection of sets. The trace of the Gaussian Free Field on S 1 was then expressed as H(x) = W (x + H), x S 1 where (14) H = {(x, y) H 1/2 < x < 1/2, y > 2 tan( πx )} (15) π The geometry of the set H allowed Astala, Jones, Kupiainen, Saksman to decouple the variables: for any two intervals I, J one only needed to decouple the white noise coming from the region I + H J + H which was above the intervals I and J. In our setting we will have to decouple also the contributions coming from the side regions as the vaguellettes ψ I have non-zero tails. We are successful in our endeavor because these tails decay very fast. 9. Decoupling With the notation from the previous section we fix a point z S 1 and n and define A i = A(z, ρ n ρ i n, 2 ρ n ρ i n), B i = B(z, ρ n ρ i n), B i = B(z, ρ n ρ i+ 1 2 n ). For simplicity of exposition, we will take A i to be square annuli. The annulus A i can be divided in two parts: R i,1 - contains all C I with base I for which I B i, j(i) A i, but for which j(i) B i+1 =. R i,2 = A i \ R i,1 A big modulus mod(g(a i )) is a consequence of controlled distortion in A i.

12 12 NICOLAE TECU Distortion in any C I depends on the doubling properties of the random measure ν in j(i). Let I be a dyadic interval in A i and any J = {J 1, J 2 } with J i j(i). We can write ν(j 1 ) ν(j 2 ) ν i(j 1 ) e sup( a J ψ J t J ψj 2 /2) ν i (J 2 ) e =: ν i(j 1 ) inf( a J ψ J t J ψj 2 /2) ν i (J 2 ) e k<i t i,k where ν is the measure obtained only by using the dyadic intervals which are included in B i. The inf and sup are over the ball B i. The sums in the exponets go over all dyadic J for which C J is in C \ 4B i. The random variables t i,k only contain the terms which correspond to the J in the region 4B k \ 4B k Distortion in R i,1. To control the distortion in this region we need to control the impact of the random variables a J where J A i and J B i+1. Let C i,i be the collection of these J. Let tr i,i = sup( J C i,i...) inf( J C i,i...), where the sup and inf are over θ R i,1. For each k > i let C i,k be the collection of dyadic intervals (on R) in B k \ B k+1 to which we add the dyadic intervals J that are inside B k and intersect B k+1, but are not contained in it. Let tr i,k = sup( J C i,k...) inf( J C i,k...), where the sup and inf are over θ R i,1. We may write then ν(j 1 ) ν(j 2 ) ν i,0(j 1 ) ν i,0 (J 2 ) e k<i t i,k e k i tr i,k for each J = {J 1, J 2 } which influences the distortion in the region R i,1. The random measure ν i,0 is obtained by considering only the random variables a J corresponding to the J which are subsets of R i,1 R. In the computations that will follow, we will actually conside r the two parts of R i,1 separately. However, their behavior is very similar so we allow ourselves to treat them as a unit. See figure 1. The random variables t i,k, tr i,k are supremums of Gaussian fields and they behave like usual Gaussian variables. This is expressed by the Borel-TIS inequality(see [1]) which we now give. Theorem 8. Let Y t be a centered Gaussian field intexed by the compact metric space T. Then P (sup Y t > u) Ae Bu u2 /2σT 2 (16) t T where σ T = max t T (EY 2 y ) 1/2 and A, B are constants. The random measures ν i,0 are independent of one another. While we can not say that with high probability the distortion in the regions R i,1

13 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 13 Figure 1. The decoupled variables used to control distortion in R i,1 is bounded (this was the case in [1]), we will construct a stopping time region inside R i,1 where the distortion grows in a controlled fashion Distortion in R i,2. Let I i be the set of I D such that C I intersects A i and j(i) B i. The distortion in R i,2 is the same as the distortion in all C I for I I i. This is a finite (and controlled) number of intervals. For each of them we only need to control 2 10 pairs J. We have already decoupled the influence of variables a J with J ouside of B i on the measure ν. For each pair J we can write (following [1]) δ νi (J) = δ νi (J 1 \ B i, J 2 \ B i) + ν i(j 1 B i) ν i (J 2 \ B i ) + ν i(j 2 B i) ν i (J 1 \ B i ) (17) ν i (J j B i) = N n k=i+1 ν i (J j B k 1 \ B k) (18) Define L i,i = δ νi (J 1 \ B i, J 2 \ B i)(19) (J 1,J 2 ) j(i),i I i ν i (J 1 L i,k = (B k 1 \ B k )) ν i (J 2 \ B (J 1,J 2 ) j(i),i I i ) + (1 2) for i + 1 k N i (20) i

14 14 NICOLAE TECU An upper bound on k i L i,k means the distortion in region R i,2 is bounded. Before we give a distributional inequality for these random variables, we need to decouple them one more time. ν i (J 1 (B k 1 \ B k )) ν i (J 2 \ B i ) ν i,k(j 1 (B k 1 \ B k )) ν i,k (J 2 \ B i ) e sup( J...) inf( J...) (21) where measure ν i,k is constructed using only the random variables a J for J 2B i and J B k. In addition, the sup and inf are considered over the set R (2B i \B k ). Denote these variables by j>k s i,k,j, where s i,j,k involves only the dyadic intervals J inside B j 1 \ B j. For simplicity of notation, we will use L i,k for the sums above, but involving ν i,k. Distortion will then be bounded by L i,k e j>k s i,k,j (22) k i For a picture of the decoupled variables see Figure 2. Figure 2. The decoupled variables L i,k and s i,k,j used to control distortion in R i,1 The random variables s i,k,j have similar properties as t i,k, tr i,k above due to the same reason: they are supremums of Gaussian fields. The random variables L i,k satisfy the following distributional inequality.

15 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 15 Lemma 9. There exists a n > 0 (depending only on the sequence of variances in the GFF) and C < (independent of i, m, ρ) such that P (L i,k > λ) Cλ 1 ρ (k i)(1+an) n In addition, L i,k and L j,l are independent if k < j or l < i. Recall that we are trying to prove an estimate on moduli of annuli at scales between ρ n ρ Nn n and ρ n. All the work we have done decoupling the distortion was done to address a fixed n. The distributional inequality is satisfied by all L i,k, where i k N n. The constant a n depends on first variances in the definition on the measures ν i,k. For each n, all the first variances are equal to the same value t n. As n increases, a n 0. This is a big difference betweek the critical case and the non critical case treated in [1]. In the sub-critical case the variables L i,k were defined for all scales k > i and they all shared the same value of the constant a. 10. Stopping time We need to control the distortion in the regions R i,2. In [1] the authors were able to get bounded distortion with high probability. In our case this is not possible anymore essentially because the variances t n t c. To control the distortion we use a stopping time algorithm. We devise a collection of rules which we apply to dyadic squares. If all these rules are satisfied for a particular interval/square, then the distortion will be controlled. Our goal is to obtain an infinite d-ary surviving tree where the distortion is controlled. In the following A l, B are large constants, δ a small constant and N is a large positive integer. We construct stopping rules on a tree in which each node represents an interval of size 2 i(n 4). For each interval I, a node in this tree, we denote by Ans n (I) the ancestor n levels above (and of size 2 (i n)(n 4) ). We start by considering a dyadic interval I of size 2 i(n 4) and measure ν which is constructed only using the vaguelets corresponding to intervals J which are subsets of I and its closest four neighbors (two to the left, two to the right - call this set J (I)). Out of all the dyadic subintervals of j(i) of size I 2 Nj we select and mark half in an alternating fashion and call them I j k. The interval I survives if all of the following good events take place: 1) 1 A i I 1 k ν(i 1 k) A i I 1 k, k (23)

16 16 NICOLAE TECU 2) For each J {I, I l, I r } ( l and r stand for left and right neighbours) I j k J ν(i j k ) A i J 2 jδ, j > 1 (24) 3) For each n i: sup 4) θ J (I) J J (Ans n((i))\j (I) sup θ J (I) J J (Ans n((i))\j (I) where θ I is the center of I. sup θ J (I) inf θ J (I) a J ψ J (θ) a J ψ J (θ I ) B2 n (25) a J ψ J (θ) a J ψ J (θ I ) B2 n (26) a J ψ J (θ) a J ψ J (θ I ) B (27) J j(i), J Ik 1 25 a J ψ J (θ) a J ψ J (θ I ) B (28) J j(i), J Ik 1 25 where θ I is the center of I. All the constants are chosen such that P (rules hold) 1 (29) The constants A l vary with the level at which we apply the rules. As we apply these rules to smaller and smaller intervals I, the properties of the measure ν become weaker and weaker. We want to make sure that the probability (29) doesn t change as we go deeper and deeper. If the all these rules are satisfied the distortion inside j(i) between heights Ik 1 25 and 2 n = 2 i(n 4) = I is bounded by D n. We reiterate the fact that this sequence won t be bounded. If all the rules are satisfied for I, we then look at its children. We consider all the dyadic intervals of size Ik 1 24 inside the region j(i). We don t have control over the distortion in the boxes corresponding to these intervals. We select one third of these intervals in such a way that any two selected intervals are separated by four boxes which we do not select. See Figure 3. There will be a total of 2 N 4 /5 such intervals. We now run the rules for each one of these intervals. When doing this, we consider the measure ν constructed only using the vaguelets corresponding to J (this interval). As we run these rules for smaller and smaller intervals we obtain a random tree. We have the following

17 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 17 Figure 3. The interval in the middle is I. The black intervals are the marked Ik 1. The green boxes represent the area where the distortion is under control. At the next step we only look at the white boxes that lie between four red boxes (picture shows only two for reasons of space). Lemma 10. P (there is a good d-ary surviving subtree) > 1 ɛ By good we mean that the distortion is controlled. A d-ary subtree is a tree where each node has d surviving children. 11. Modulus estimate In this section we are concerned with a deterministic modulus estimate. Assume we have a mapping G : C C with distortion µ and an annulus A centered on the real axis(for simplicity we will use a dyadic annulus of radii 1 and 2). Assume µ = 0 in the lower half plane. Assume the distortion in the top part of the annulus is bounded by D and that in the two sides we have connected sub-domains Ω 1 and Ω 2 of A. Ω j is constructed in a stopping time fashion on a dyadic grid. At level 2 Ni the number of surviving intervals is d i and the distortion there is bounded by D i. One should think of Ω j converging to a Cantor set E j on R A. See figure 4. The next theorem is a generalization of the following statement: if G is conformal in the top of A and inside Ω 1 Ω 2 then mod(f (A)) > c 0 which depends on the logarithmic capacity of the sets E 1, E 2. Theorem 11. Mod(G(A)) > α = const. Proof. One way to prove this result is to construct two closed curves inside G(A) which are neither too long, nor too close to one another.

18 18 NICOLAE TECU Figure 4. The annulus A and the mapping G. The black regions are the ones where we have no control on the distortion. When G is conformal (as above) this is a consequence of Pfluger s theorem (see for example [4] for a reference) and a modulus estimate. We will use a more direct argument here. We know that modg(a) = 1, where Γ is the family of curves which connect the two components of the complement of G(A). We want to obtain an upper bound modg( Γ) on modg( Γ) and we do this by constructing a good metric in G(A). u(z) G z G z We consider the metric ρ on G(A): ρ(w) = where G(z) = w and u is defined below. Then modg( Γ) Ω ρ2 da(w) G(Ω) u 2 D(z)dA(z), where D(z) is the distortion of G. For this argument to work we need to take u such that u dz γ 1 for all γ Γ. In Ω 2 we take u(x, y) = ( x f 1 y(t)dt, y) and similarly for Ω 1. For each y the density f y (t) puts all the mass uniformly on the surviving intervals defining Ω j and zero in the regions where we have no control on the distortion. We may say that for y [2 Ni, 2 Ni+N ], f y (t) = 2 Ni d i on the intervals defining Ω 2. Set u = 1 at all points where the distortion is bounded (the lower half of A and the top of A). Now we have u dz u dz 1 so u is an admissible γ γ metric.

19 RANDOM CONFORMAL WELDING TOWARDS CRITICALITY 19 Now we have that C + C i mod(g( Γ)) C C i N(i 1) 2 2 Ni 1 f 2 y (x)d(x, y)dxdy + C f 2 y (x)d i (x)dxdy + C 2 Ni D i 2 2Ni d 2i 2 Ni d i = C + C i D i d i So if D i increases slow enough the modulus will be bounded by a constant. 12. Main probabilistic estimate revisited We are now in a position to give an outline of the proof of the main probabilistic estimate. Fix n and z. ρ n is fixed. We look of for ρ n, N n, b n and c n such that (13) holds. If all of the following events hold for a large number of annuli t i,k, tr i,k are small There is an infinite d-ary surviving subtree in region R i,1. L i,k, s i,k,j are small, the sum of the moduli is big. Small sum of moduli implies a large number of failures for at least one of these rules. We have probabilistic estimates for each such failure. We will treat each rule separately and due to the decoupling we will be able to use the independence of variables corresponding to some of the scales. References [1] Astala K., Jones P.W., Kupiainen A., Saksman E. Random conformal weldings preprint [2] Astala, K., Iwaniec T., Martin G., Elliptic partial differential equatons and quasiconformal mappings in the plane, Princeton University Press (2009). [3] Bacry, E., Muzy, J.F. Log-infinitely divisible multifractal processes Comm. Math. Physics. 236, 2003, [4] Balogh, Z., Bonk, M. Lengths of radii under conformal maps of the unit disc Proc. Amer. Math. Soc. 127, 3, , (1999) [5] Donoho, David, L. Nonlinear solution of linear inverse problems by waveletvaguelette decomposition Applied and Computational Harmonic Analysis 2, , (1995). [6] Duplantier, B., Sheffield, S. Liouville quantum gravity and KPZ Invent. math. 185 (2011), [7] Jones, Peter W. and Smirnov, Stanislav S., Removability theorems for Sobolev functions and quasiconformal maps Ark. Mat. 38 (2000), ;

20 20 NICOLAE TECU [8] Kahane, J.-P., Sur le chaos multiplicatif Ann. Sci. Math. Quebec 9, 1985, [9] Kahane, J.-P., Peyriere, J. Sur certaines martingales de Benoit Mandelbrot Advances in Math. 22, 1976, [10] Koskela, Pekka; Nieminen, Tomi Quasiconformal removability and the quasihyperbolic metric Indiana Univ. Math. J. 54 (2005), no. 1, [11] Lawler, G. F. Conformally invariant processes in the plane AMS (2005) [12] Lawler, G. F., Schramm, O., Werner W.Conformal invariance of planar looperased random walks and uniform spanning trees Annals of Prob., 32, [13] Mandelbrot, B. B. Intermittent turbulence in self-similar cascades:divergence of high moments and dimension of carrier Journal of Fluid Mechanics 62, 1974, [14] Mandelbrot, B. B., Fisher, A., Calvet, L. The Multifractal Model of Asset Returns Cowles Foundation discussion paper no. 1164, Yale University, paper available from the SSRN database at [15] Meyer, Y. Ondelettes et Operateurs Herman Editeurs des sciences et des arts., [16] Robert, R., Vargas, C. Gaussian multiplicative chaos revisited ArXiv [math.pr] , [17] Schramm, O. Scaling limits of loop-erased random walks and uniform spanning trees Israel J. Math. 118 (2000), [18] Schramm, O., Sheffield, S. Contour lines of the two dimensional discrete Gaussian free field Acta Math, 202(1), 2009, [19] Schramm, O., Sheffield, S. A contour line of the continuum Gaussian free field Arxiv e-prints, 2010, [20] Sheffield, S. Gaussian free fields for mathematicians Probab. Theory Related Fields, 139(3-4), 2007, [21] Sheffield, S. Conformal weldings of random surfaces: SLE and the quantum gravity zipper, arxiv: v1 (2010) [22] Smirnov, S. Critical percolation in the plane:conformal invariance, Cardy s formula, scaling limits C.R. Acad.Sci.Paris S. I Math. 333, no. 3, [23] Werner, W. Percolation et modele d Ising, Societe Mathmatique de France, 2009