What is a random surface?

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 the Brownian sphere, the peanosphere, the pure Liouville quantum gravity sphere, or a certain conformal field theory. All of these objects have concise definitions, and are all in some sense equivalent, but the equivalence is highly non-trivial, building on hundreds of math and physics papers over the past half century. More generally, the “continuum random surface embedded in d-dimensional Euclidean space” makes a kind of sense for d ∈ (−∞, 25) even when d is not a positive integer; and this can be extended to higher genus surfaces, surfaces with boundary, and surfaces with marked points or other decoration. These constructions have deep roots in both mathematics and physics, drawing from classical graph theory, complex analysis, probability and representation theory, as well as string theory, planar statistical physics, random matrix theory and a simple model for twodimensional quantum gravity. We present here an informal, colloquium-level overview of the subject, which we hope will be accessible to both newcomers and experts. We aim to answer, as cleanly as possible, the fundamental question. What is a random surface? © 2022 International Mathematical Union To be Published by EMS Press. Proc. Int. Cong.Math. 2022. ar X iv :2 20 3. 02 47 0v 1 [ m at h. PR ] 4 M ar 2 02 2 As a random path, Brownian motion is canonical in the sense that it is uniquely characterized by certain symmetries, and universal in the sense that it is a limit of many kinds of discrete random walks. This paper will describe a similarly canonical and universal random surface. This random surface has several different formulations. To tell the story in a fanciful way, imagine a dialog, first about random paths, then about random surfaces. INSTRUCTOR: Consider the simple random walk on Z. At each time step a coin toss decides whether position goes up or down. If you shrink the graph horizontally by a factor of C and vertically by a factor of √ C, then the C → ∞ limit is Brownian motion. 20 40 60 80 100 5 10 15 STUDENT: Great! But can you define Brownian motion directly in the continuum? INSTRUCTOR: Sure! Fix 0 = t0<t1< . < tn. Specify the joint law of B(t1), . , B(tn) by making increments B(tk )−B(tk−1) independent normals with mean 0, variance tk−tk−1. Extend to countable dense set (Kolmogorov extension), then all t (Kolomogorov-Čentsov). STUDENT: Are there other natural ways to characterize Brownian motion? INSTRUCTOR: Brownian motion is canonical in that it is the only random path with certain symmetries (like stationarity/independence of increments). It is universal in that (per central limit theorem) it is a limit of many discrete walks. It comes up everywhere. STUDENT: What if I want a random path embedded in R? INSTRUCTOR: Use a vector ( B1 (t), B2 (t), . , Bd (t) ) of independent Brownian motions. The student is happy. Now imagine a similar dialog for random surfaces. INSTRUCTOR: Take a uniformly random triangulation of the sphere with n triangles. The picture below is an example with 30,000 triangles by Budzinski, using some software to embed the surface in three dimensions and give us a view. The n → ∞ limit of this object is a random fractal surface called the Brownian sphere. It is also a peanosphere and a pure Liouville quantum gravity sphere and a conformal field theory.