In a wire, electrons rebound off each other in such a complicated fashion that there’s no way to follow exactly what’s happening.

But over the last 50 years, mathematicians and physicists have begun to grasp that this blizzard of movement settles into elegant statistical patterns. Electron movement takes one statistical shape in a conductor and a different statistical shape in an insulator.

That, at least, has been the hunch. Over the last half-century mathematicians have been searching for mathematical models that bear it out. They’ve been trying to prove that this beautiful statistical picture really does hold absolutely.

And in a paper posted online last summer, a trio of mathematicians have come the closest yet to doing so. In that work, Paul Bourgade of New York University, Horng-Tzer Yau of Harvard University, and Jun Yin of the University of California, Los Angeles, prove the existence of a mathematical signature called “universality” that certifies that a material conducts electricity.

“What they show, which I think is a breakthrough mathematically … is that first you have conduction, and second [you have] universality,” said Tom Spencer, a mathematician at the Institute for Advanced Study in Princeton, New Jersey.

The paper is the latest validation of a grand vision for quantum physics set forward in the 1960s by the famed physicist Eugene Wigner. Wigner understood that quantum interactions are too complicated to be described exactly, but he hoped the essence of those interactions would emerge in broad statistical strokes.

This new work establishes that, to an extent that might have surprised even Wigner, his hope was well-founded.

Universally Strange

Even seemingly isolated and unrelated events can fall into a predictable statistical pattern. Take the act of murder, for example. The stew of circumstances and emotions that combine to lead one person to kill another is unique to each crime. And yet someone observing crime statistics in the heat of an urban summer can predict with a high degree of accuracy when the next body will fall.

There are many different types of statistical patterns that independent events can follow. The most famous statistical pattern of all is the normal distribution, which takes the shape of a bell curve and describes the statistical distribution of a wide range of uncorrelated events (like heights in a population or scores on the SAT). There’s also Zipf’s law, which describes the relative sizes of the largest numbers in a data set, and Benford’s law, which characterizes the distribution of first digits in the numbers in a data set.

In the 1950s, Wigner confronted a problem and needed the help of a new statistical pattern to solve it. More than a decade after he’d helped instigate the Manhattan Project, he wanted to model interactions between the hundreds of particles inside the uranium nucleus. The problem was too complicated to tackle directly.

“A large nucleus is a complicated thing; we have no idea how to understand it from first principles,” Spencer said.

So Wigner simplified the problem: He ignored individual particle interactions, which were too hard to map, and instead focused on the average statistical behavior of the whole system, which was more tractable.

Wigner implemented this picture using a grid of numbers that specify how particles interact. This grid is known as a matrix. It’s like a technical appendix for the Schrödinger equation, which is the equation used to describe the behavior of subatomic particles. By specifying the numbers in the matrix exactly, you specify the interactions exactly.

Wigner couldn’t do that, so instead he filled the matrix with random numbers. He hoped this simplification would enable him to proceed with his calculations, while still producing a useful description of the uranium nucleus at the end.

Which it did. Wigner found that he was able to extract a pattern from his “random” matrix. The pattern involved a second layer of numbers called eigenvalues, which are like the DNA of a matrix. Puzzlingly, his random matrix had correlated eigenvalues. On a number line, the eigenvalues seemed to exhibit a somewhat regular spacing — never clustered together nor spread too far apart. It was almost as if they were magnets, pushing each other toward an even spacing. The resulting distribution is now often referred to as the Wigner-Dyson-Mehta distribution (after the three physicists who contributed to its discovery). It describes a phenomenon called universality.

To get a sense of universality, consider how tall people are. In the real world, if you started plucking people two at a time from a crowd in Times Square, there’s a reasonable chance that you’d find pairs of people with approximately the same height. But if heights in a population followed the Wigner-Dyson-Mehta distribution, you wouldn’t expect two randomly selected people to have similar heights at all. The heights would be correlated in such a way that the first person’s height was always different from the second person’s.

Universality describes many different kinds of things: the frequency and size of avalanches, the timing of buses in decentralized transit systems, and even the spacing of cells in the retina of a chicken. It pertains, in general, to complex, correlated systems.