# What is a random surface?

@inproceedings{Sheffield2022WhatIA, title={What is a random surface?}, author={Scott Sheffield}, year={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 topological sphere. As n → ∞ these random surfaces (appropriately scaled) converge in law. The limit is a “canonical” sphere-homeomorphic random surface, much the way Brownian motion is a canonical random path. Depending on how the surface space and convergence topology are specified, the limit is…

## 3 Citations

### Liouville quantum gravity from random matrix dynamics

- Mathematics
- 2022

We establish the ﬁrst connection between 2 d Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if ( U t ) is a Brownian motion on the unitary group at…

### 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…

### Quantum Gravity in 30 Questions

- Physics
- 2022

Quantum gravity is the missing piece in our understanding of the fundamental interactions today. Given recent observational breakthroughs in gravity, providing a quantum theory for what lies beyond…

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