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Uniqueness of the welding problem for SLE and LQG

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RGMW06 - RGM follow up

Fix $\kappa \in (0,8)$ and suppose that $\eta$ is an SLE $\kappa$ curve in $\mathbb{H}$ from $0$ to $\infty$. We show that if $\varphi \colon \mathbb{H} \to \mathbb{H}$ is a homeomorphism which is conformal on $\mathbb{H} \setminus \eta$ and $\varphi(\eta)$, $\eta$ are equal in distribution then $\varphi$ is a conformal automorphism of $\mathbb{H}$. Applying this result for $\kappa=4$ establishes that the welding operation for critical ($\gamma=2$) Liouville quantum gravity (LQG) is well-defined. Applying it for $\kappa \in (4,8)$ gives a new proof that the welding of two looptrees of quantum disks to produce an SLE $\kappa$ on top of an independent $4/\sqrt{\kappa}$-LQG surface is well-defined. These results are special cases of a more general uniqueness result which applies to any non-space-filling SLE -type curve (e.g., the exotic SLE $_\kappa^\beta(\rho)$ processes). This is a joint work with Oliver McEnteggart and Jason Miller.

This talk is part of the Isaac Newton Institute Seminar Series series.

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