10th

**Universality: In Mysterious Pattern, Math and Nature Converge**

"In 1999, while sitting at a bus stop in Cuernavaca, Mexico, a Czech physicist named Petr Šeba noticed young men handing slips of paper to the bus drivers in exchange for cash. It wasn’t organized crime, he learned, but another shadow trade: Each driver paid a “spy” to record when the bus ahead of his had departed the stop. If it had left recently, he would slow down, letting passengers accumulate at the next stop. If it had departed long ago, he sped up to keep other buses from passing him. This system maximized profits for the drivers. And it gave Šeba an idea. (…)

The interaction between drivers caused the spacing between departures to exhibit a distinctive pattern previously observed in quantum physics experiments. (…) “We felt here some kind of similarity with quantum chaotic systems.” (…) A “spy” network makes the decentralized bus system more efficient. As a consequence, the departure times of buses exhibit a ubiquitous pattern known as “universality.” (…)

Subatomic particles have little to do with decentralized bus systems. But in the years since the odd coupling was discovered, the same pattern has turned up in other unrelated settings. Scientists now believe **the widespread phenomenon, known as “universality,**” stems from an underlying connection to mathematics, and it is helping them to model complex systems from the internet to Earth’s climate. (…)

The red pattern exhibits **a precise balance of randomness and regularity known as “universality,”** which has been observed in the spectra of many complex, correlated systems. In this spectrum, **a mathematical formula called the “correlation function” gives the exact probability of finding two lines spaced a given distance apart. **(…)

The pattern was first discovered in nature in the 1950s in the energy spectrum of the uranium nucleus, a behemoth with hundreds of moving parts that quivers and stretches in infinitely many ways, producing an endless sequence of energy levels. In 1972, the number theorist Hugh Montgomery observed it in the zeros of the Riemann zeta function, a mathematical object closely related to the distribution of prime numbers. In 2000, Krbálek and Šeba reported it in the Cuernavaca bus system. And in recent years it has shown up in spectral measurements of composite materials, such as sea ice and human bones, and in signal dynamics of the Erdös–Rényi model, a simplified version of the internet named for Paul Erdös and Alfréd Rényi. (…)

**Each of these systems has a spectrum — a sequence like a bar code representing data such as energy levels, zeta zeros, bus departure times or signal speeds. In all the spectra, the same distinctive pattern appears: The data seem haphazardly distributed, and yet neighboring lines repel one another, lending a degree of regularity to their spacing. This fine balance between chaos and order, which is defined by a precise formula, also appears in a purely mathematical setting: It defines the spacing between the eigenvalues, or solutions, of a vast matrix filled with random numbers**. (…)

It seems to be a law of nature,” said Van Vu, a mathematician at Yale University who, with Terence Tao of the University of California, Los Angeles, **has proven universality for a broad class of random matrices**.**Universality is thought to arise when a system is very complex, consisting of many parts that strongly interact with each other to generate a spectrum. The pattern emerges in the spectrum of a random matrix, for example, because the matrix elements all enter into the calculation of that spectrum. But random matrices are merely “toy systems” that are of interest because they can be rigorously studied, while also being rich enough to model real-world systems**, Vu said. Universality is much more widespread. Wigner’s hypothesis (named after Eugene Wigner, the physicist who discovered universality in atomic spectra) asserts that all complex, correlated systems exhibit universality, from a crystal lattice to the internet.

**The more complex a system is, the more robust its universality should be**, said László Erdös of the University of Munich, one of Yau’s collaborators. **“This is because we believe that universality is the typical behavior.”**

— Natalie Wolchover, *In Mysterious Pattern, Math and Nature Converge*, Wired, Feb 6, 2013. (Photo: Marco de Leija)

See also:

☞ Mathematics of Disordered Quantum Systems and Matrices, IST Austria.