Scott Sheffield: What is a random surface?

We will survey the modern theory of “random surfaces” while also reviewing the rich history of the subject and presenting numerous computer illustrations and animations. There are many ways to begin, but one is to consider a finite collection of unit equilateral triangles. There are finitely many ways to glue each edge to a partner, and we obtain a random sphere-homeomorphic surface by sampling uniformly from the gluings that produce a topological sphere. As the number of triangles tends to infinity, 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, the bosonic string 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 any d ∈ (−∞,25]. This story can also 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 two-dimensional quantum gravity.
We present here a colloquium-level overview of the subject, which we hope will be accessible to both newcomers and experts. We aim to answer, as simply and cleanly as possible, the fundamental question. What is a random surface?
Slides: www.mathunion....