Conformal welding of independent Gaussian multiplicative chaos measures

TL;DR

The paper proves that two independent Gaussian multiplicative chaos boundary measures on the unit circle can be conformally welded for sufficiently small GMC parameters $\gamma$, yielding conformal maps $f_1,f_2$ with $f_1\circ\phi_1^{-1}=f_2\circ\phi_2^{-1}$ and uniqueness up to Möbius transformations. The authors tackle a degenerate Beltrami equation via Lehto-type approximation, establishing boundary Hölder continuity by controlling image annuli through a detailed probabilistic analysis of GMC measures and an oscillating random-walk algorithm that locates many suitable annuli. A novel approximate decomposition of GMC measures using hyperbolic white noise underpins scale-separated estimates, enabling high-probability control of shape and size events across scales. The results are extended to general subcritical GMC parameters and to simultaneous welding of rotated homeomorphisms, connecting to SLE/LQG welding literature while providing an alternative, measure-driven welding framework with potential broader applicability.

Abstract

We solve the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian multiplicative chaos measures with small parameter values. In other words, given two such measures on the boundary of the unit disk we show that there exist conformal maps to complementary domains on the Riemann sphere such that the pushforward of the normalised measures agree on their common boundary.

Figures (12)

• Figure 1: The conformal maps $f_1,f_2$ can be chosen so that $f_1(1)=f_2(1)$ and for $x,y\in[0,1]$ we have $f_1(e^{2\pi ix})=f_2(e^{2\pi i y})$ if and only if $\tau^{(1)}([0,x])/\tau^{(1)}([0,1])=\tau^{(2)}([0,y])/\tau^{(2)}([0,1])$. Crosses and circles denote points to be matched up in this way. The images of $\partial\mathbb{D}$ under $f_1$ and $f_2$ coincide to give a closed curve in $\mathbb{C}$ which we think of as a subset of $\hat{\mathbb{C}}\simeq\mathbb{S}^2$.
• Figure 2: The sets $\mathcal{H}$ and $\mathcal{H}_\epsilon$ are used to define a white noise representation for the Gaussian free field.
• Figure 3: To bound the modulus of continuity of a homeomorphism $g$ near a point $z_0$, it is enough to show that the images of small annuli $\mathbb{A}_i$ surrounding $x$ are not too 'distorted' (i.e., they are not too long or thin). We apply this reasoning to $g=F_n$ around points $z_0\in\partial\mathbb{D}$.
• Figure 4: The half-annuli $A_t(x)$ and $A_{t+1}(x)$.
• Figure 5: The images of the upper and lower half-annuli $A_t(x)$ and $\widetilde{A}_s(y)$ may 'match up', in the sense that their union contains a topological annulus, if $\Psi_1(x)\approx\Psi_2(y)$. This is equivalent to finding annuli around points of $\partial\mathbb{D}$, although it is notationally easier to work on $\mathbb{R}$.

Theorems & Definitions (75)

• Lemma 1.1: ajks
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof