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In Mysterious Pattern, Math and Nature Converge

By investigating the “universality” phenomenon in random matrices, researchers have developed a better sense of why it arises elsewhere — and how it can be used. In a flurry of recent papers, Yau and other mathematicians have characterized many new types of random matrices, which can conform to a variety of numerical distributions and symmetry rules. For example, the numbers filling a matrix’s rows and columns might be chosen from a bell curve of possible values, or they might simply be 1s and –1s. The top right and bottom left halves of the matrix might be mirror images of one another, or not. Time and again, regardless of their specific characteristics, the random matrices are found to exhibit that same chaotic yet regular pattern in the distribution of their eigenvalues. That’s why mathematicians call the phenomenon “universality.”





In many simple systems, individual components can assert too great an influence on the outcome of the system, changing the spectral pattern. With larger systems, no single component dominates. “It’s like if you have a room with a lot of people and they decide to do something, the personality of one person isn’t that important,” Vu said.

Whenever a system exhibits universality, the behavior acts as a signature certifying that the system is complex and correlated enough to be treated like a random matrix. “This means you can use a random matrix to model it,” Vu said. “You can compute other parameters of the matrix model and use them to predict that the system may behave like the parameters you computed.”

This technique is enabling scientists to understand the structure and evolution of the internet. Certain properties of this vast computer network, such as the typical size of a cluster of computers, can be closely estimated by measurable properties of the corresponding random matrix. “People are very interested in clusters and their locations, partially motivated by practical purposes such as advertising,” Vu said.

A similar technique may lead to improvements in climate change models. Scientists have found that the presence of universality in features similar to the energy spectrum of a material indicates that its components are highly connected, and that it will therefore conduct fluids, electricity or heat. Conversely, the absence of universality may show that a material is sparse and acts as an insulator. In new work presented in January at the Joint Mathematics Meetings in San Diego, Ken Golden, a mathematician at the University of Utah, and his student, Ben Murphy, used this distinction to predict heat transfer and fluid flow in sea ice, both at the microscopic level and through patchworks of Arctic melt ponds spanning thousands of kilometers.

When Arctic melt ponds are sufficiently connected, as pictured here, they exhibit a property called universality that researchers believe is common to all complex, correlated systems. (Photo: Don Perovich)

The spectral measure of a mosaic of melt ponds, taken from a helicopter, or a similar measurement taken of a sample of sea ice in an ice core, instantly exposes the state of either system. “Fluid flow through sea ice governs or mediates very important processes that you need to understand in order to understand the climate system,” Golden said. “The transitions in the eigenvalue statistics presents a brand new, mathematically rigorous approach to incorporating sea ice into climate models.”

The same trick may also eventually provide an easy test for osteoporosis. Golden, Murphy and their colleagues have found that the spectrum of a dense, healthy bone exhibits universality, while that of a porous, osteoporotic bone does not.

“We’re dealing with systems where the ‘particles’ can be on the millimeter or even on the kilometer scale,” Murphy said, referring to the systems’ component parts. “It’s amazing that the same underlying mathematics describes both.”

The reason a real-world system would exhibit the same spectral behavior as a random matrix may be easiest to understand in the case of the nucleus of a heavy atom. All quantum systems, including atoms, are governed by the rules of mathematics, and specifically by those of matrices. “That’s what quantum mechanics is all about,” said Freeman Dyson, a retired mathematical physicist who helped develop random matrix theory in the 1960s and 1970s while at Princeton’s Institute for Advanced Study. “Every quantum system is governed by a matrix representing the total energy of the system, and the eigenvalues of the matrix are the energy levels of the quantum system.”

The matrices behind simple atoms, such as hydrogen or helium, can be worked out exactly, yielding eigenvalues that correspond with stunning precision to the measured energy levels of the atoms. But the matrices corresponding to more complex quantum systems, such as a uranium nucleus, quickly grow too thorny to grasp. According to Dyson, this is why such nuclei can be compared to random matrices. Many of the interactions inside uranium — the elements of its unknown matrix — are so complex that they become washed out, like a mélange of sounds blending into noise. Consequently, the unknown matrix that governs the nucleus behaves like a matrix filled with random numbers, and so its spectrum exhibits universality.

Scientists have yet to develop an intuitive understanding of why this particular random-yet-regular pattern, and not some other pattern, emerges for complex systems. “We only know it from calculations,” Vu said. Another mystery is what it has to do with the Riemann zeta function, whose spectrum of zeros exhibits universality. The zeros of the zeta function are closely tied to the distribution of the prime numbers — the irreducible integers out of which all others are constructed. Mathematicians have long wondered at the haphazard way in which the primes are sprinkled along the number line from one to infinity, and universality offers a clue. Some think there may be a matrix underlying the Riemann zeta function that is complex and correlated enough to exhibit universality. Discovering such a matrix would have “big implications” for finally understanding the distribution of the primes, said Paul Bourgade, a mathematician at Harvard.

Or perhaps the explanation lies deeper still. “It may happen that it is not a matrix that lies at the core of both Wigner’s universality and the zeta function, but some other, yet undiscovered, mathematical structure,” Erdös said. “Wigner matrices and zeta functions may then just be different representations of this structure.”

Many mathematicians are searching for the answer, with no guarantee that there is one. “Nobody imagined that the buses in Cuernavaca would turn out to be an example of this. Nobody imagined that the zeroes of the zeta function would be another example,” Dyson said. “The beauty of science is it’s completely unpredictable, and so everything useful comes out of surprises.”