Random Conformal Weldings

TL;DR

This paper constructs a conformally invariant random family of closed curves in the plane using welding of random homeomorphisms derived from the exponential of a free field, employing probabilistic estimates and Beltrami equations.

Contribution

It introduces a novel method to generate conformally invariant random curves via welding of homeomorphisms based on free fields, extending the understanding of random conformal geometry.

Findings

Constructed a new family of random conformal curves.

Proved existence using Lehto's method and probabilistic estimates.

Suggested a connection to SLE curves for future research.

Abstract

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle. The homeomorphism is constructed using the exponential of $\beta X$ where $X$ is the restriction of the two dimensional free field on the circle and the parameter $\beta$ is in the "high temperature" regime $\beta<\sqrt 2$. The welding problem is solved by studying a non-uniformly elliptic Beltrami equation with a random complex dilatation. For the existence a method of Lehto is used. This requires sharp probabilistic estimates to control conformal moduli of annuli and they are proven by decomposing the free field as a sum of independent fixed scale fields and controlling the correlations of the complex dilation restricted to dyadic cells of various scales. For uniqueness we invoke a result by Jones and Smirnov on conformal removability of H\"older curves. We conjecture that our curves are locally related to SLE$(\kappa)$ for $\kappa<4$.

Figures

Equations

$$\mathbb{E}X(z)X(z^{\prime })=-\log |z-z^{\prime }|$$

$$"d\tau =e^{\beta X(z)}dz".$$

$$h(\theta )=\tau ([0,\theta ))/\tau ([0,1))\text{mbox}for\theta \in [0,1).$$

$$x\sim y,\text{mbox}whenx\in \mathbb{T}=\partial \mathbb{D},\text{mbox}y=\varphi (x)\in \mathbb{T}=\partial {\mathbb{D}}_{\infty }.$$

$$f_{+}:\mathbb{D}\to {\Omega }_{+}\text{mbox}and{f}_{-}:{\mathbb{D}}_{\infty }\to {\Omega }_{-}$$

$$\varphi ={\left(f_{+}\right)}^{-1}\circ f_{-}$$

$$\frac{\partial f}{\partial \overline{z}}=\mu (z)\frac{\partial f}{\partial z},\text{mbox}foralmosteveryz\in \Omega ,$$

$$\varphi =f{|}_{\mathbb{T}},$$

$$\frac{\partial F}{\partial \overline{z}}=\chi (z)\mu (z)\frac{\partial F}{\partial z},\text{mbox}fora.e.z\in \mathbb{C}.$$

$$f_{-}:{\mathbb{D}}_{\infty }\to {\Omega }_{-}$$

$$F(z)=f_{+}\circ f(z),z\in \mathbb{D},$$

$$\varphi (z)=f{|}_{\mathbb{T}}(z)={\left(f_{+}\right)}^{-1}\circ f_{-}(z),z\in \mathbb{T}.$$

$$K(\varphi ):=\underset{s,t\in \mathbb{R}}{\sup }\frac{|\varphi (e^{2\pi i(s+t)})-\varphi (e^{2\pi is})|}{|\varphi (e^{2\pi i(s-t)})-\varphi (e^{2\pi is})|}<\infty .$$

$$K(z)=\frac{1+|\mu (z)|}{1-|\mu (z)|},z\in \Omega ,$$

$$A(z,r,R):=\{w\in \mathbb{C}:r<|w-z|<R\}.$$

$$L(z,r,R):=L_K(z,r,R):={\int }_r^R\frac{1}{{\int }_0^{2\pi }K\left(z+\rho e^{i\theta }\right)d\theta }\frac{d\rho }{\rho }$$

$$K\in L_{loc}^1(\mathbb{C}),$$

$$L_K(z,0,R_0)=\infty ,\text{mbox}forallz\in \mathbb{C}.$$

$$\frac{\partial f}{\partial\overline{z}}(z)=\mu(z)\frac{\partial f}{\partial z}(z)\quad\mbox{ for almost every $z\in\mathbb{C},$}$$

$$K(z)\in L_{loc}^{\infty }(\mathbb{D}).$$

$$\varphi (z)={\left(f_{+}\right)}^{-1}\circ f_{-}(z),z\in \mathbb{T}.$$

$$\frac{\partial F}{\partial \overline{z}}=\chi (z)\mu (z)\frac{\partial F}{\partial z},\text{mbox}fora.e.z\in \mathbb{C}.$$

$$K(z)=\frac{1+|\chi (z)\mu (z)|}{1-|\chi (z)\mu (z)|},$$

$$F(z)=f_{+}\circ f(z),z\in \mathbb{D},$$

$$\frac{D_O(f(A(w,r,R)))}{D_I(f(A(w,r,R)))}\ge \frac{1}{16}\exp \left(2{\pi }^2{L}_{K_f}(w,r,R)\right).$$

$$mod(A)=2\pi \log \frac{R}{r}$$

$$mod(f(A(w,r,R)))\ge 2\pi L_{K_f}(w,r,R).$$

$$\frac{1}{16}\exp (\pi mod(A))\le \frac{D_O(A)}{D_I(A)}.$$

$$\varphi (z)={\left(f_{+}\right)}^{-1}\circ f_{-}(z),z\in \mathbb{T},$$

$$g_{\pm }=\Phi \circ f_{\pm },\Phi :\hat{\mathbb{C}}\to \hat{\mathbb{C}}\text{mbox}M\ddot{o}bius.$$

Full text

1. Introduction

There has been great interest in conformally invariant random curves and fractals in the plane ever since it was realized that such geometric objects appear naturally in statistical mechanics models at the critical temperature [8]. A major breakthrough in the field occurred when O. Schramm [28] introduced the Schramm-Loewner Evolution (SLE), a stochastic process whose sample paths are conjectured (and in several cases proved) to be the curves occurring in the physical models. We refer to [29] and [30] for a general overview and some recent work on SLE. SLE curves come in two varieties: the radial one where the curve joins a boundary point (say of the disc) to an interior point and the chordal case where two boundary points are joined.

SLE describes a curve growing in time: the original curve of interest (say a cluster boundary in a spin system) is obtained as time tends to infinity. In this paper we give a different construction of random curves which is stationary i.e. the probability measure on curves is directly defined without introducing an auxiliary 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. Consider a Jordan curve $\gamma$ bounding a simply connected region $\Omega$ in the plane. By the Riemann mapping theorem there are conformal maps $f_{\pm}$ mapping the unit disc $\mathbb{D}$ and its complement to $\Omega$ and its complement. The map $f_{+}^{-1}\circ f_{-}$ extends continuously to the boundary ${\mathbb{T}}=\partial\mathbb{D}$ of the disc and defines a homeomorphism of the circle. Conformal Welding is the inverse operation where, given a suitable homeomorphism of the circle, one constructs a Jordan curve on the plane (see Section 2). In fact, in our case the curve is determined up to a Möbius transformation of the plane. Thus random curves (modulo Möbius transformations) can be obtained from random homeomorphisms via welding.

In this paper we introduce a random scale invariant set of homeomorphisms $h_{\omega}:{\mathbb{T}}\to{\mathbb{T}}$ and construct the welding curves. The model considered here has been proposed by the second author. The construction depends on a real parameter $\beta$ (”inverse temperature”) and the maps are a.s. in $\omega$ Hölder continuous for $\beta<\beta_{c}$. For this range of $\beta$ the welding map will be a.s. well-defined. For$\beta>\beta_{c}$ we expect the map $h_{\omega}$ not to be continuous and no welding to exist. We conjecture that the resulting curves should locally ”look like” SLE$(\kappa)$ for $\kappa<4$ but we don’t have good arguments for this. The case $\beta=\beta_{c}$, presumably corresponding to SLE$(4)$, is not covered by our analysis.

Since we are interested in random curves that are self similar it is natural to take $h$ with such properties. Our choice for $h$ is constructed by starting with the Gaussian random field $X$ on the circle (see Section 3 for precise definitions) with covariance

[TABLE]

where $z,z^{\prime}\in\mathbb{C}$ with modulus one. $X$ is just the restriction of the 2d massless free field (Gaussian Free Field) on the circle. The exponential of $\beta X$ gives rise to a random measure $\tau$ on the unit circle ${\mathbb{T}}$, formally given by

[TABLE]

The proper definition involves a limiting process $\tau(dz)=\lim_{\varepsilon\to 0}e^{\beta X_{\varepsilon}(z)}/{{\mathbb{E}}\,}e^{\beta X_{\varepsilon}(z)}dz$, where $X_{\varepsilon}$ stands for a suitable regularization of $X,$ see Section 3.3 below.

Identifying the circle as ${\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}=[0,1)$ our random homeomorphism $h:[0,1)\to[0,1)$ is defined as

[TABLE]

The main result of this paper can then be summarized as follows:

For $\beta^{2}<2$ and almost surely in $\omega$, the formula (3) defines a Hölder continuous circle homeomorphism, such that the welding problem has a solution $\gamma$, where $\gamma$ is a Jordan curve bounding a domain $\Omega=f_{+}(\mathbb{D})$ with a Hölder continuous Riemann mapping $f_{+}$. For a given $\omega$, the solution is unique up to a Möbius map of the plane.

We refer to Section 5 (Theorem 5.2) for the exact statement of the main result.

Apart from connection to SLE the weldings constructed in this paper should be of interest to complex analysts as they form a natural family that degenerates as $\beta\uparrow\sqrt{2}.$ It would be of great interest to understand the critical case $\beta=\sqrt{2}$ as well as the low temperature ”spin glass phase” $\beta>\sqrt{2}.$ It would also be of interest to understand the connection of our weldings to those arising from stochastic flows [1]. In [1] Hölder continuous homeomorphisms are considered, but the boundary behaviour of the welding and hence its existence and uniqueness are left open.

In writing the paper we have tried be generous in providing details on both the function theoretic and the stochastics notions and tools needed, in order to serve readers with varied backgrounds. The structure of the paper is as follows. Section 2 contains background material on conformal welding and the geometric analysis tools we need later on. To be more specific, Section 2 recalls the notion of conformal welding and explains how the welding problem is reduced to the study of the Beltrami equation. Also we recall a useful method due to Lehto [21] to prove the existence of a solution for a class of non-uniformly elliptic Beltrami equations, and a theorem by Jones and Smirnov [15] that will be used to verify the uniqueness of our welding. Finally we recall the Beurling-Ahlfors extension of homeomorphisms of the circle to the inside the disc. For our purposes we need to estimate carefully the dependence of the dilatation of the extension in a Whitney cube by just using small amount of information of the homeomorphism on a ’shadow’ of the cube.

In Section 3 we introduce the one-dimensional trace of the Gaussian free field and recall some known properties of its exponential that we will use to define and study the random circle homeomorphism. Section 4 is the technical core of the paper as it contains the main probabilistic estimates we need to control the random dilatation of the extension map. Finally, in Section 5 things are put together and the a.s. existence and uniqueness of the welding map is proven.

Let us finish by a remark on notation. We denote by $c$ and $C$ generic constants which may vary between estimates. When the constants depend on parameters such as $\beta$ we denote this by $C(\beta)$.

Acknowledgements. We thank M. Bauer, D. Bernard, Steffen Rohde and Stanislav Smirnov for useful discussions. This work is partially funded by the Academy of Finland, European Research Council and National Science Foundation.

2. Conformal Welding

In the present section we recall for the general readers benefit basic notions and results from geometric analysis that are needed in our work. In particular, we recall the notion of conformal welding, Lehto’s method for solving the Beltrami-equations, the uniqueness result for weldings due to Jones and Smirnov, and the last subsection contains estimates for the Beurling-Ahlfors extension tailored for our needs.

2.1. Welding and Beltrami equation

One of the main methods for constructing conformally invariant families of (Jordan) curves comes from the theory of conformal welding. Put briefly, in this method we glue the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ and the exterior disk $\mathbb{D}_{\infty}=\{z\in\widehat{\mathbb{C}}:|z|>1\}$ along a homeomorphism $\phi:{\mathbb{T}}\to{\mathbb{T}}$, by the identification

[TABLE]

The problem of welding is to give a natural complex structure to this topological sphere. Uniformizing the complex structure then gives us the curve, the image of the unit circle.

More concretely, given a Jordan curve $\Gamma\subset\widehat{\mathbb{C}}$, let

[TABLE]

be a choice of Riemann mappings onto the components of the complement $\widehat{\mathbb{C}}\setminus\Gamma=\Omega_{+}\cup\Omega_{-}$. By Caratheodory’s theorem $f_{-}$ and $f_{+}$ both extend continuously to $\partial\mathbb{D}=\partial\mathbb{D}_{\infty}$, and thus

[TABLE]

is a homeomorphism of ${\mathbb{T}}$. In the welding problem we are asked to invert this process; given a homeomorphism $\phi:{\mathbb{T}}\to{\mathbb{T}}$ we are to find a Jordan curve $\Gamma$ and conformal mappings $f_{\pm}$ onto the complementary domains $\Omega_{\pm}$ so that (4) holds.

It is clear that the welding problem, when solvable, has natural conformal invariance attached to it; any image of the curve $\Gamma$ under a Möbius transformation of $\widehat{\mathbb{C}}$ is equally a welding curve. Similarly, if $\phi:{\mathbb{T}}\to{\mathbb{T}}$ admits a welding, then so do all its compositions with Möbius transformations of the disk. Note, however, that not all circle homeomorphisms admit a welding, for examples see [24] and [32].

The most powerful tool in solving the welding problem is given by the Beltrami differential equation, defined in a domain $\Omega$ by

[TABLE]

where we look for homeomorphic solutions $f\in W^{1,1}_{loc}(\Omega)$. Here (5) is an elliptic system whenever $|\mu(z)|<1$ almost everywhere, and uniformly elliptic if there is a constant $0\leq k<1$ such that $\|\mu\|_{\infty}\leq k$.

In the uniformly elliptic case, homeomorphic solutions to (5) exist for every coefficient with $\|\mu\|_{\infty}<1$, and they are unique up to post-composing with a conformal mapping [4, p.179]. In fact, it is this uniqueness property that gives us a way to produce the welding. To see this suppose first that

[TABLE]

where $f\in W^{1,2}_{loc}(\mathbb{D};\mathbb{D})\cap C(\overline{\mathbb{D}})$ is a homeomorphic solution to (5) in the disc $\mathbb{D}$. Find then a homeomorphic solution to the auxiliary equation

[TABLE]

Now $\Gamma=F({\mathbb{T}})$ is a Jordan curve. Moreover, as $\partial_{\overline{z}}F=0$ for $|z|>1$, we can set $f_{-}:=F|_{\mathbb{D}_{\infty}}\mbox{ and }\Omega_{-}:=F(\mathbb{D}_{\infty})$ to define a conformal mapping

[TABLE]

On the other hand, since both $f$ and $F$ solve the equation (5) in the unit disk $\mathbb{D}$, by uniqueness of the solutions we have

[TABLE]

for some conformal mapping $f_{+}:\mathbb{D}=f(\mathbb{D})\to\Omega_{+}:=F(\mathbb{D})$. Finally, on the unit circle,

[TABLE]

Thus we have found a solution to the welding problem, under the assumption (6). That the welding curve $\Gamma$ is unique up to a Möbius transformation of $\mathbb{C}$ follows from [4, Theorem 5.10.1], see also Corollary 2.5 below.

To complete this circle of ideas we need to identify the homeomorphisms $\phi:{\mathbb{T}}\to{\mathbb{T}}$ that admit the representation (5), (6) with uniformly elliptic $\mu$ in (5). It turns out [4, Lemma 3.11.3 and Theorem 5.8.1] that such $\phi$ ’s are precisely the quasisymmetric mappings of ${\mathbb{T}}$, mappings that satisfy

[TABLE]

2.2. Existence in the degenerate case: the Lehto condition

The previous subsection describes an obvious model for constructing random Jordan curves, by first finding random homeomorphisms of the circle and then solving for each of them the associated welding problem. In the present work, however, we are faced with the obstruction that circle homeomorphisms with derivative the exponentiated Gaussian free field almost surely do not satisfy the quasisymmetry assumption (10). Thus we are forced outside the uniformly elliptic PDE’s and need to study (5) with degenerate coefficients with only $|\mu(z)|<1$ almost everywhere. We are even outside the much studied class of maps of exponentially integrable distortion, see [4, 20.4.] In such generality, however, the homeomorphic solutions to (5) may fail to exist, or the crucial uniqueness properties of (5) may similarly fail.

In his important work [21] Lehto gave a very general condition in the degenerate setting, for the existence of homeomorphic solutions to (5). To recall his result, assume we are given the complex dilatation $\mu=\mu(z)$, and write then

[TABLE]

for the associated distortion function. Note that $K(z)$ is bounded precisely when the equation (5) is uniformly elliptic, i.e. $\|\mu\|_{\infty}<1$. Thus the question is how strongly can $K(z)$ grow for the basic properties of (5) still to remain true. In order to state Lehto’s condition we fix some notation. For given $z\in{\mathbb{C}}$ and radii $0\leq r<R<\infty$ let us denote the corresponding annulus by

[TABLE]

In the Lehto approach one needs to control the conformal moduli of image annuli in a suitable way. This is done by introducing for any annulus $A(w,r,R)$ and for the given distortion function $K$ the following quantity, which we call the Lehto integral:

[TABLE]

For the following formulation of Lehto’s theorem see [4, p. 584].

Theorem 2.1**.**

Suppose $\mu$ is measurable and compactly supported with $|\mu(z)|<1$ for almost every $z\in\mathbb{C}$. Assume that the distortion function $K(z)=(1+|\mu(z)|)/(1-|\mu(z)|)$ is locally integrable,

[TABLE]

and that for some $R_{0}>0$ the Lehto integral satisfies

[TABLE]

Then the Beltrami equation

[TABLE]

admits a homeomorphic $W^{1,1}_{loc}$-solution $f:\mathbb{C}\to\mathbb{C}$.

Corollary 2.2**.**

Suppose $\phi:{\mathbb{T}}\to{\mathbb{T}}$ extends to a homeomorphism $f:\mathbb{C}\to\mathbb{C}$ satisfying (12) - (14) together with the condition

[TABLE]

Then $\phi$ admits a welding: there are a Jordan curve $\Gamma\subset\widehat{\mathbb{C}}$ and conformal mappings $f_{\pm}$ onto the complementary domains of $\Gamma$ such that

[TABLE]

**Proof. ** Given the extension $f:\mathbb{C}\to\mathbb{C}$ let us again look at the auxiliary equation

[TABLE]

Since Lehto’s condition holds as well for the new distortion function

[TABLE]

we see from Theorem 2.1 that the auxiliary equation (16) admits a homeomorphic solution $F:\mathbb{C}\to\mathbb{C}$. Arguing as in (6) - (9) it will be then sufficient to show that

[TABLE]

where $f_{+}$ is conformal in $\mathbb{D}$. But this is a local question; every point $z\in\mathbb{D}$ has a neighborhood where $K(z)$ is uniformly bounded, by (15). In such a neighborhood the usual uniqueness results to solutions of (5) apply, see [4, p.179]. Thus $f_{+}$ is holomorphic, and as a homeomorphism it is conformal. This proves the claim.

Consequently, in the study of random circle homeomorphisms $\phi=\phi_{\omega}$ a key step for the conformal welding of $\phi_{\omega}$ will be to show that almost surely each such mapping admits a homeomorphic extension to $\mathbb{C}$, where the distortion function satisfies a condition such as (13). In our setting where derivative of $\phi$ is given by the exponentiated trace of a Gaussian free field, the extension procedure is described in Section 2.4 and the appropriate estimates it requires are proven in Section 4.

Actually, in Section 5 when proving our main theorem we need to present a variant of Lehto’s argument where it will be enough to estimate the Lehto integral only at a suitable countable set of points $z\in{\mathbb{T}}$. We also utilize there the fact that the extension of our random circle homeomorphism $\phi$ satisfies (15). In verifying the Hölder continuity of the ensuing map we shall apply a useful estimate (Lemma 2.3 below) that estimates the geometric distortion of an annulus under a quasiconformal map.

Given a bounded (topological) annulus $A\subset{\mathbb{C}}$ , with $E$ the bounded component of ${\mathbb{C}}\setminus A,$ we denote by $D_{O}(A):={{\rm diam}\,}(A)$ the outer diameter, and by $D_{I}(A):={{\rm diam}\,}(E)$ the inner diameter of A.

Lemma 2.3**.**

Let $f$ be a quasiconformal mapping on the annulus $A(w,r,R)$, with the distortion function $K_{f}.$ It then holds that

[TABLE]

**Proof. ** Recall first that for a rigid annulus $A=A(w,r,R)$ the modulus

[TABLE]

while for any topological annulus $A$, we define its conformal modulus by $mod(A)=mod(g(A))$ where $g$ is a conformal map of $A$ onto a rigid annulus. Then we have [4, Cor. 20.9.2] the following basic estimate for the modulus of the image annulus in terms of the Lehto integrals:

[TABLE]

On the other hand, by combining [33, 7.38 and 7.39] and [3, 5.68(16)] we obtain for any bounded topological annulus $A\subset{\mathbb{C}}$

[TABLE]

Put together, the desired estimate follows.

2.3. Uniqueness of the welding

An important issue of the welding is its uniqueness, that the curve $\Gamma$ is unique up to composing with a Möbius transformation of $\widehat{\mathbb{C}}$. As the above argument indicates, this is essentially equivalent to the uniqueness of solutions to the appropriate Beltrami equations, up to a Möbius transformation. However, in general the assumptions of Theorem 2.1 alone are much too weak to imply this.

It fact, in our case the uniqueness of solutions to the Beltrami equation (16) is equivalent to the conformal removability of the curve $F({\mathbb{T}})$. Recall that a compact set $B\subset\widehat{\mathbb{C}}$ is conformally removable if every homeomorphism of $\widehat{\mathbb{C}}$ which is conformal off $B$ is conformal in the whole sphere, hence a Möbius transformation.

It follows easily that e.g. images of circles under quasiconformal mappings, i.e. homeomorphisms satisfying (19) with $\|\mu\|_{\infty}<1$, are conformally removable, while Jordan curves of positive area are never conformally removable.

For general curves the removability is a deep problem; no characterizations of conformally removable Jordan curves is known to this date. What saves us in the present work is that we have available the remarkable result of Jones and Smirnov in [15]. We will not need their result in its full generality, as the following special case will sufficient for our purposes.

Theorem 2.4**.**

(Jones, Smirnov [15]) Let $\Omega\subset\widehat{\mathbb{C}}$