Random Geometry

Random is often the word chosen to describe something that has no order. But randomness has become an increasingly useful tool in mathematics, a discipline whose meaningfulness relies, primarily, on order.

In statistics, randomness as a measure of uncertainty, makes possible the identification of events, whether sociopolitical or physical, with the use of probability distributions. We use random sampling to create tools that reduce our uncertainty about whether something has actually happened or not. In information theory, entropy quantifies uncertainty, and makes the analysis of information, in the broadest sense, possible. In algorithmic information theory randomness helps us quantify complexity.  Randomness characterizes the emergence of certain kinds of fractals found in nature and even the action of organisms. It has been used to explore neural networks, both natural and artificial. Researchers, for example, have explored the use of a chaotic system in a machine that might then have properties important to brain-like learning, adaptability and flexibility.  Gregory Chaitin’s metabiology, outlined in his book Proving Darwin: Making Biology Mathematical, investigates the random evolution of artificial software that might provide insight into the random evolution of natural software (DNA).

Quanta Magazine recently published a piece with the title, A Unified Theory of Randomness, in which Kevin Hartnett describes the work of MIT professor of mathematics Scott Sheffield, who investigates the properties of shapes that are created by random processes. These are shapes that occur naturally in the world but, until now, appeared to have only their randomness in common.

Yet in work over the past few years, Sheffield and his frequent collaborator, Jason Miller, a professor at the University of Cambridge, have shown that these random shapes can be categorized into various classes, that these classes have distinct properties of their own, and that some kinds of random objects have surprisingly clear connections with other kinds of random objects. Their work forms the beginning of a unified theory of geometric randomness.

“You take the most natural objects — trees, paths, surfaces — and you show they’re all related to each other,” Sheffield said. “And once you have these relationships, you can prove all sorts of new theorems you couldn’t prove before.”

In the coming months, Sheffield and Miller will publish the final part of a three-paper series that for the first time provides a comprehensive view of random two-dimensional surfaces — an achievement not unlike the Euclidean mapping of the plane.

The article is fairly thorough in making the meaning of these advances accessible. But with limited time and space, I’ll just highlight a few things:

In this ‘random geometry,’ if the location of some of the points of a randomly generated object are known, probabilities are assigned to subsequent points. As it turns out, certain probability measures arise in many different contexts.  This contributes to the identification of classes and properties, critical to growth in mathematics.

We can all imagine random motion, or random paths, but here the random surface is explored. As Hartnett tells us,

Brownian motion is the “scaling limit” of random walks — if you consider a random walk where each step size is very small, and the amount of time between steps is also very small, these random paths look more and more like Brownian motion. It’s the shape that almost all random walks converge to over time.

Two-dimensional random spaces, in contrast, first preoccupied physicists as they tried to understand the structure of the universe.

Sheffield was interested in finding a Brownian motion for surfaces. And two ideas that already existed would help lead him.  Physicists have a way of describing a random surface, whose surface area could be determined (related to quantum gravity). There is also something called a Brownian map, whose structure allows the calculation of distance between points. But the two could not be shown to be related. If there was a way to measure distance on former structure, it could be compared to distances measured on the latter.  Their hunch was that these two surfaces were different perspectives on the same object.  To overcome the difficulty of distance measurement on the former, they used growth over time as a distance metric.

…as Sheffield and Miller were soon to learn, “[random growth] becomes easier to understand on a random surface than on a smooth surface,” said Sheffield. The randomness in the growth model speaks, in a sense, the same language as the randomness on the surface on which the growth model proceeds. “You add a crazy growth model on a crazy surface, but somehow in some ways it actually makes your life better,” he said.

But they needed another trick to model growth on very random surfaces in order to establish a distance structure equivalent to the one on the (very random) Brownian map.  They found it in a curve.

Sheffield and Miller’s clever trick is based on a special type of random one-dimensional curve that is similar to the random walk except that it never crosses itself. Physicists had encountered these kinds of curves for a long time in situations where, for instance, they were studying the boundary between clusters of particles with positive and negative spin (the boundary line between the clusters of particles is a one-dimensional path that never crosses itself and takes shape randomly). They knew these kinds of random, noncrossing paths occurred in nature, just as Robert Brown had observed that random crossing paths occurred in nature, but they didn’t know how to think about them in any kind of precise way. In 1999 Oded Schramm, who at the time was at Microsoft Research in Redmond, Washington, introduced the SLE curve (for Schramm-Loewner evolution) as the canonical noncrossing random curve.

Popular opinion often finds fault in attempts to quantify everything, as if quantification is necessarily diminishing of things.  What strikes me today is that quantification is more the means to finding structure.  But it is the integrity of those structures that consistently unearths surprises.  The work described here is a beautiful blend of ideas that bring new depth to the value of geometric perspectives.

“It’s like you’re in a mountain with three different caves. One has iron, one has gold, one has copper — suddenly you find a way to link all three of these caves together,” said Sheffield. “Now you have all these different elements you can build things with and can combine them to produce all sorts of things you couldn’t build before.”

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