# Integrability of SLE via conformal welding of random surfaces

@inproceedings{Ang2021IntegrabilityOS, title={Integrability of SLE via conformal welding of random surfaces}, author={Morris Ang and Nina Holden and Xin Sun}, year={2021} }

We demonstrate how to obtain integrability results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called SLEκ(ρ−; ρ+). Moreover, we prove that the SLE loop measure constructed by Zhan (2020) arises naturally from the conformal welding of two quantum disks in LQG. Our proofs are…

## 10 Citations

### What is a random surface?

- Mathematics
- 2022

Given 2n unit equilateral triangles, there are finitely many ways to glue each edge to a partner. We obtain a random sphere-homeomorphic surface by sampling uniformly from the gluings that produce a…

### Brownian half-plane excursion and critical Liouville quantum gravity

- Mathematics
- 2022

In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be obtained by gluing together a…

### A deterministic approach to Loewner-energy minimizers

- Mathematics
- 2022

. We study two minimization questions: the nature of curves γ ⊂ H which minimize the Loewner energy among all curves from 0 to a ﬁxed z 0 ∈ H , and the nature of γ which minimize the Loewner energy…

### The SLE loop via conformal welding of quantum disks

- Mathematics
- 2022

. We prove that the SLE loop measure arises naturally from the conformal welding of two Liouville quantum gravity (LQG) disks for γ 2 = κ ∈ (0 , 4). The proof relies on our companion work on…

### On Natural Measures of Several Random Fractals Induced from SLE and CLE

- Mathematics
- 2022

In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal…

### On Natural Measures of SLE- and CLE-Related Random Fractals

- Mathematics
- 2022

In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal…

### The moduli of annuli in random conformal geometry

- Mathematics, Physics
- 2022

. We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The ﬁrst is for the law of the modulus of the Brownian annulus describing…

### Baxter permuton and Liouville quantum gravity

- Mathematics
- 2022

The Baxter permuton is a random probability measure on the unit square which describes the scaling limit of uniform Baxter permutations. We ﬁnd an explict formula for the expectation of the Baxter…

### Integrability of the conformal loop ensemble

- Mathematics
- 2021

We demonstrate that the conformal loop ensemble (CLE) has a rich integrable structure by establishing exact formulas for two CLE observables. The first describes the joint moments of the conformal…

### FZZ formula of boundary Liouville CFT via conformal welding

- Mathematics
- 2021

Liouville Conformal Field Theory (LCFT) on the disk describes the conformal factor of the quantum disk, which is the natural random surface in Liouville quantum gravity with disk topology. Fateev,…

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