• Corpus ID: 247244552

# 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

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## References

SHOWING 1-10 OF 255 REFERENCES

### Random geometry on the sphere

We introduce and study a universal model of random geometry in two dimensions. To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class

### The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to Liouville quantum gravity

• Mathematics
• 2020
Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a √ 8/3-Liouville

### The Brownian map is the scaling limit of uniform random plane quadrangulations

We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n−1/4, converge as n → ∞ in distribution for the Gromov–Hausdorff

### The topological structure of scaling limits of large planar maps

We discuss scaling limits of large bipartite planar maps. If p≥2 is a fixed integer, we consider, for every integer n≥2, a random planar map Mn which is uniformly distributed over the set of all

### The Brownian Map

NinaHoldenwon the 2021MaryamMirzakhaniNew Frontiers Prize for her work on random surfaces and the mathematics of quantum gravity in two dimensions. Here I will explain just one concept involved in

### Scaling limits of random bipartite planar maps with a prescribed degree sequence

This paper studies the asymptotic behaviour of uniform random maps with a prescribed face-degree sequence, in the bipartite case, and shows that, properly rescaled, such maps converge in distribution towards the Brownian map in the Gromov-Hausdorff sense.

### An axiomatic characterization of the Brownian map

• Mathematics
Journal de l’École polytechnique — Mathématiques
• 2015
The Brownian map is a random sphere-homeomorphic metric measure space obtained by "gluing together" the continuum trees described by the $x$ and $y$ coordinates of the Brownian snake. We present an

### Random Surfaces and Liouville Quantum Gravity

Liouville quantum gravity (LQG) surfaces are a family of random fractal surfaces which can be thought of as the canonical models of random two-dimensional Riemannian manifolds, in the same sense that

### Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

• Mathematics
Electronic Journal of Probability
• 2021
Recent works have shown that random triangulations decorated by critical ($p=1/2$) Bernoulli site percolation converge in the scaling limit to a $\sqrt{8/3}$-Liouville quantum gravity (LQG) surface

### Liouville quantum gravity spheres as matings of finite-diameter trees

• Physics
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
• 2019
We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure