Mysterious structures unite mathematics and nature

In the city of Cuernavaca in Mexico, a “spy” network improves the efficiency of the bus fleet. As a result, the departure schedule of buses everywhere matches the pattern of "universality"

In 1999, while sitting at a bus stop in the city of Cuernavaca in Mexico, Czech physicist Peter Sheba [Petr Šeba] noticed people giving paper pieces to the bus driver in exchange for money. He discovered that it was not a manifestation of organized crime, but another “shadow” trade: each driver paid a “spy” who noted when the previous bus had left the stop. If he left recently, the driver of this bus slowed down so that passengers could get together at the next stop. If that bus departed long ago, the driver accelerated so that other buses would not overtake him. Such a system maximized the profits of drivers. Which gave Sheba an idea.

“We thought we were seeing a situation somewhat resembling chaotic quantum systems,” explained the co-author of Sheba, Milan Krbálek [Milan Krbálek].

After several unsuccessful attempts to communicate with the "spies", Sheba asked his student to explain that he was not from the tax or from the mafia. He is just a mad scientist changing tequila to their data. And people gave him their notes. When the researchers plotted thousands of bus times, their suspicions were confirmed. The interaction between drivers led to the distribution of gaps between transport waste, which coincides with the structure of some experiments in quantum physics.
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“I thought that something like this might appear, but I was surprised to see such an exact coincidence,” said Sheba.

Subatomic particles have little in common with a decentralized bus system. But over the years since the discovery of strange quantum interactions, the same data structures have appeared in other unrelated situations. Scientists believe that this is a common phenomenon, known as "universality", is due to the mathematical connection of phenomena, and helps them simulate complex systems, from the Internet to the planet's climate.


The red graph represents the perfect balance between randomness and periodicity, known as universality. It is observed in the spectra of many complex systems with correlation. In this spectrum, the mathematical correlation function gives the exact probability of having two lines at a given distance from each other.

Such a structure was first found in nature in the 1950s in the energy spectrum of the uranium core , a monster with hundreds of moving parts, oscillating and stretching in an infinite number of ways, and producing an endless sequence of energy levels. In 1972, number theory specialist Hugh Montgomery observed her in the zeros of the Riemann zeta function , a mathematical object related to the distribution of primes. In 2000, Krbalek and Sheba discovered her in a bus departure schedule in Cuernavaca . And recently, it appeared in the spectral measurements of composite materials, such as sea ice and human bone tissue, and in the dynamics of signals from the Erds – Rényi model , a simplified version of the Internet.

Each of these systems has a spectrum — a sequence-bar code representing data such as energy levels, zetas zeros, bus departure times, or speed signals. Identical structures appear throughout the spectrum. The distribution of data looks random, but at the same time adjacent lines “repel each other” from each other, which leads to a certain regularity of the intervals. The exact balance between chaos and order, defined by a formula, also appears in purely mathematical problems: it determines the distance between the eigenvalues ​​of a matrix filled with random numbers.

“Why so many physical systems behave like random matrices remains a mystery,” says Horng Tzer Yao, a mathematician at Harvard University. “But in the last three years we have taken a very important step to understand this.”

To study the phenomenon of universality in random matrices, scientists have understood a little about why it appears everywhere, and how it can be used. In the pile of new works, Yao and other mathematicians described many new types of random matrices that obey several numerical distributions and symmetry rules. For example, the numbers for columns and rows of matrices can be taken from the normal distribution curve of possible values, or you can fill it with values ​​of 1 and -1. The upper right and lower left parts of the matrix may or may not mirror each other. And regardless of their characteristics, random matrices exhibit the same chaotic, but regular spectra in the distribution of their eigenvalues. Therefore, mathematicians called this phenomenon “universality”.

“It looks like a law of nature,” says Van Wu, a mathematician at Yale University, who, together with Terence Tao of the University of California at Los Angeles, proved versatility for a wide class of random matrices.

It is believed that universality appears in very complex systems consisting of many parts, closely interacting with each other to create a spectrum. The configuration appears in the spectrum of a random matrix, for example, because all elements of the matrix are used in the calculation of this spectrum. But random matrices, according to Wu, are just “toy systems,” they are simple enough to learn, and rich enough to simulate real systems. Versatility is much more common. Wigner's hypothesis (named after Eugene Wigner, the discovery that opened up universality in the spectrum of atoms) suggests that all complex systems with correlation are versatile, from the crystal lattice to the Internet.

The more complex the system, the more versatile it manifests itself, says Laszlo Erdös of the University of Munich, one of Yao’s work colleagues. "You believe that universality is typical behavior."

In many simple systems, individual components may too much influence the overall result of the work, which changes the appearance of the spectrum. Larger systems do not dominate any single component. “It turns out that there is a room with a large number of people deciding to do something, and the identity of one of them is not so important,” says Wu.


Mathematicians use random matrices to study and predict some properties of the Internet, for example, the size of a typical computer cluster.

When the system demonstrates universality, it guarantees that it is complex and there is enough correlation inside it so that it can be interpreted as a random matrix. “This means that you can use a random matrix to model it,” says Wu. "You can calculate other parameters of the model based on the matrix and use them to predict the behavior of this system."

This technique allows scientists to understand the structure and evolution of the Internet. Some properties of this huge computer network, such as the typical size of a cluster of computers, can be fairly accurately estimated through the measured properties of the corresponding random matrix. “People are interested in clusters and their location, and this is often due to practical goals, such as advertising, for example,” says Wu.

Similar technologies may lead to improvements in climate change models. Scientists have found that the presence of universality, similar to the energy spectrum, in the material indicates a strong coherence of its parts, and, as a consequence, good conductivity of liquids, electricity or heat. And conversely, the lack of universality can speak of the sparsity of the material and its insulating properties. A new paper presented at the San Diego Math Conference, Ken Golden, a mathematician from the University of Utah, and his student, Ben Murphy, used this distinction to predict heat conduction and fluid flow in sea ice, both at the microscopic and arctic polynyas present in areas stretching for thousands of kilometers.

The spectral measurement of a mosaic of melted wormwood, photographed from a helicopter, or sea ice data obtained from a sample, shows the state of each of these systems. “The flow of fluid through the sea ice drives very important processes that need to be understood in order to understand the operation of the climate system,” says Golden. “Transitions in statistics of own solutions represent a new, mathematically rigorous approach to the inclusion of sea ice in climate models.”

The same trick can lead to the creation of a simple test for osteoporosis. Golden, Murphy, and their colleagues found that the spectrum of dense, healthy bone is universal, while the spectrum of porous is not.


Arctic polynyas are distinguished by their universality if they have sufficient coherence.

“We work with systems whose“ particles ”can be millimeter or kilometer-sized,” says Murphy about system components. “It's amazing that the same math describes them all.”

The reason why real systems have similar behavior with a random matrix is ​​probably the easiest to understand in the case of a heavy atom nucleus. All quantum systems, including atoms, operate according to the rules of mathematics, especially with the participation of matrices. “This is the essence of quantum mechanics,” said Freeman Dyson, a former mathematician who helped develop the theory of random matrices in the 1960s and 1970s at the Princeton Institute for Advanced Development. “Each quantum system is described by a matrix representing its total energy, and the matrix’s own solutions are the energy levels of the quantum system.”

The matrices of simple atoms, hydrogen and helium, can be calculated exactly, and the obtained own solutions coincide with surprising accuracy with the measured energy levels of the atoms. But matrices of more complex systems, such as the nucleus of uranium, become too "prickly" to "grasp" them. According to Dyson, because of this, such kernels can be compared with a random matrix. Many interactions inside uranium — elements of an unknown matrix — are so complex that their mixture produces noise, like a multitude of superimposed sounds. As a result, the unknown matrix controlling the kernel behaves like a matrix with random numbers, and their spectrum is universal.


Such unrelated wormwood has no universality, their spectrum is random.

Scientists have not yet developed an intuitive understanding of why complex systems demonstrate this and not another random / periodic sequence. “We only know this from calculations,” says Wu. Another mystery is their connection with the Riemann zeta function, in which universality is manifested in the spectrum of zeros. These zeros are closely related to the distribution of primes — non-convertible integers, of which all others consist. Mathematicians have long sought a description of the distribution of primes along the number line from 1 to infinity, and universality gives them the key. Some believe that a matrix that is sufficiently complex and coherent for the possession of universality can be behind the Riemann zeta function. The discovery of such a matrix would have a "strong influence" on the understanding of the distribution of primes, as Paul Bourgade, a Harvard mathematician, said.

It is possible that the explanation is hidden even deeper. “It may turn out that the center of Wigner's universality and zeta functions is not a matrix, but some kind of not yet open mathematical structure,” says Erds. “The Wigner matrices and the zeta functions may turn out to be different representations of this structure.”

Many mathematicians are looking for an answer with no guarantee of its existence. “No one imagined that buses in Cuernavaca would be such an example. No one imagined that the zeros of the zeta function would be another example, says Dyson. “The beauty of science is its unpredictability, and therefore everything useful comes out of surprises.”