The Brownian Map

Nina Holden won the 2021 Maryam Mirzakhani New Frontiers Prize for her work on random surfaces and the mathematics of quantum gravity. I’d like to tell you what she did… but I’m so far behind I’ll just explain a bit of the background.

Suppose you randomly choose a triangulation of the sphere with n triangles. This is a purely combinatorial thing, but you can think of it as a metric space if each of the triangles is equilateral with all sides of length 1.

This is a distorted picture of what you might get, drawn by Jérémie Bettinelli:

The triangles are not drawn as equilateral, so we can fit this shape into 3d space. Visit Bettinelli’s page for images that you can rotate:

• Jérémie Bettinelli, Computer simulations of random maps.

I’ve described how to build a random space out of triangles. In the limit if you rescale the resulting space by a factor of so it doesn’t get bigger and bigger, it converges to a ‘random metric space’ with fascinating properties. It’s called the Brownian map.

This random metric space is on average so wrinkly and crinkly that ‘almost surely’—that is, with probability 1—its Hausdorff dimension is not 2 but 4. And yet it is almost surely homeomorphic to a sphere!

Rigorously proving this is hard: a mix of combinatorics, probability theory and geometry.

Ideas from physics are also important here. There’s a theory called ‘Liouville quantum gravity’ that describes these random 2-dimensional surfaces. So, physicists have ways of—nonrigorously—figuring out answers to some questions faster than the mathematicians!

A key step in understanding the Brownian map was this paper from 2013:

• Jean-François Le Gall, Uniqueness and universality of the Brownian map, Annals of Probability 41 (2013), 2880–2960.

The Brownian map is to surfaces what Brownian motion is to curves. For example, the Hausdorff dimension of Brownian motion is almost surely 2: twice the dimension of a smooth curve. For the Brownian map it’s almost surely 4, twice the dimension of a smooth surface.

Let me just say one more technical thing. There’s a ‘space of all compact metric spaces’, and the Brownian map is actually a probability measure on this space! It’s called the Gromov-Hausdorff space, and it itself is a metric space… but not compact. (So no, we don’t have a set containing itself as an element.)

There’s a lot more to say about this… but I haven’t gotten very close to understanding Nina Holden’s work yet. She wrote a 7-paper series leading up to this one:

• Nina Holden and Xin Sun, Convergence of uniform triangulations under the Cardy embedding.

They show that random triangulations of a disk can be chosen to a random metric on the disk which can also be obtained from Liouville quantum gravity.

This is a much easier place to start learning this general subject:

• Ewain Gwynne, Random surfaces and Liouville quantum gravity.

One reason I find all this interesting is that when I worked on ‘spin foam models’ of quantum gravity, we were trying to develop combinatorial theories of spacetime that had nice limits as the number of discrete building blocks approached infinity. We were studying theories much more complicated than random 2-dimensional triangulations, and it quickly became clear to me how much work it would be to carefully analyze these. So it’s interesting to see how mathematicians have entered this subject—starting with a much more tractable class of theories, which are already quite hard.

While the theory I just described gives random metric spaces whose Hausdorff dimension is twice their topological dimension, Liouville quantum gravity actually contains an adjustable parameter that lets you force these metric spaces to become less wrinkled, with lower Hausdorff dimension. Taming the behavior of random triangulations gets harder in higher dimensions. Renate Loll and Jan Ambjørn have argued that we need to work with Lorentzian rather than Riemannian geometries to get physically reasonable behavior.