The matrix is a workhorse of modern math — and of physics, computer science and engineering, for that matter. This mathematical object is an array of numbers that encodes a transformation between mathematical spaces. Matrices can represent the way various elements of a system rely on one another. Among researchers who study social networks, for example, an adjacency matrix that represents the connections between people will use a 1 to indicate that two people in the network know each other and a 0 to indicate they are strangers.
In the 1950s, physicist Eugene Wigner began to use matrices to model the interactions of atomic nuclei. These interactions are so complicated, particularly in a system large enough to represent anything of real-world significance, that it is impossible to discern the exact matrices involved. Instead, he had the insight that random matrices could be used instead. The idea of a random matrix is that instead of looking at a matrix with certain fixed values, one imagines a matrix in which all values are selected randomly from a set of numbers with fixed parameters. Wigner believed these random matrices would behave in particular ways, shedding light on the systems they were modeling.
A few years ago, Simons Investigator and Harvard mathematics professor Horng-Tzer Yau and his collaborators successfully answered key questions Wigner had posed, closing that chapter in the book of random matrices. Yau didn’t start his career studying random matrices. For many years, he worked on a variety of problems related to statistical mechanics, including fluid dynamics and complicated many-body systems that arise in astrophysics. In 2000, he was awarded a MacArthur fellowship, popularly known as a ‘genius grant,’ in recognition of the contributions he had made to both mathematics and physics in studying those problems.
Shortly before starting to tackle Wigner’s conjecture, Yau had been focusing on questions related to the quantum mechanical behavior of semiconductors using random Schrödinger operators. This work built on that of Nobel Prize winner Philip Anderson, who modeled semiconductors with impurities as lattices with randomly distributed obstacles. As Yau was working on Anderson’s model, he found that he kept hearing that Wigner’s random matrices were lurking in the background. “The question was always mentioned in meetings,” he says. Eventually, he and some of his co-investigators decided to alter course and work on Wigner matrices themselves.
In order to say anything meaningful about a matrix full of random numbers — indeed, to say anything meaningful about random numbers at all — the researcher must make assumptions about how the numbers are chosen. Are the numbers distributed normally — the classic bell curve — or taken from some more exotic distribution? Not long after Wigner posed the problem, Freeman Dyson and other physicists were able to successfully probe the case where the values were distributed normally, but other distributions remained elusive. Wigner and other physicists who studied the question believed that the behavior of these large random matrices would be the same no matter the distribution from which the entries were drawn.
On its face, that was a startling claim. How could properties based on a particular random distribution turn out to be independent of that distribution? But there was a precedent for such a situation: The central limit theorem in statistics asserts an analogous result. In essence, Yau and colleagues showed that Wigner’s conjecture was a sort of central limit theorem for random matrices.
“You sometimes stumble on things in unexpected ways.”
These days, Yau is extending his work in several directions. He hopes to circle back to the work on Anderson’s model of semiconductors, eventually determining whether they really are governed, as Anderson predicted, by the same rules as random matrices. Yau sees the semiconductor as an important example to study, in part because the randomness of the system is entirely natural rather than imposed by humans. “Most random matrix examples are constructed by something like flipping a coin, a human construction,” Yau says. “The Anderson model is so precisely given by a law of nature.”
Although some of Wigner’s beliefs about random matrices have been confirmed, there are as yet no examples of physical systems that conform to those random matrices. Yau hopes to construct a quantum mechanical model that will actually display the behaviors of Wigner’s random matrices.
Whereas Wigner’s original vision for random matrices came from his work on quantum mechanics at the level of atomic nuclei, Yau’s work has taken that vision on a winding path away from quantum mechanics into pure mathematics. It may well continue into the decidedly macroscopic realms of human social networks and other forms of data analysis. “That’s how it happens in science,” Yau says. “You sometimes stumble on things in unexpected ways.”
His boldest dreams, though, are about finding applications of random matrix theory beyond the quantum world where Wigner’s work started. With computing power and dataset sizes at all-time highs, the networks available for analysis are more massive than ever. Yau hopes that the techniques he has developed in random matrix theory will offer insight into the inner workings of neural networks — computer systems that ‘learn’ by finding patterns in data. So far, even finding an appropriate mathematical language to describe what a neural network does has eluded mathematics. “Mathematicians are not even sure what the right question is yet,” he says.
On the other hand, neural networks are large, complicated correlated systems — exactly the type of systems Wigner considered. Although the precise mathematics of how neural networks are related to random matrices is unclear, Yau is hopeful that new insights from the study of random matrix theory will allow his team — or other researchers — to tackle the modern challenges in understanding neural networks and machine learning.
