Universality

Scientists have discovered a mysterious pattern that somehow connects a bus system in Mexico and chicken eyes to quantum physics and number theory. It’s known as Universality.

 

TRANSCRIPT

A hidden pattern is popping up in seemingly unrelated places -- from a bus system in Mexico, to chicken eyes, to number theory, and quantum physics.

This phenomenon, known as universality, continues to surprise mathematicians and reveal a deeper understanding of our world.

♪♪ Imagine arriving at an empty bus stop in New York City.

The last bus must have just left, but the sign says the bus comes every 10 minutes on average.

What's the probability that the next bus will arrive within five minutes?

The probability of a random event happening within some interval, like a bus coming within the next five minutes, is given by a curve called a Poisson Distribution.

But what if bus-arrival times are not independent?

In the 1970s, in Cuernavaca, Mexico, bus drivers would hire spies to sit along their route and the drivers either speed up or wait at a stop depending on how long ago their spy said the previous bus left.

This spaced out the buses and maximized their profits.

♪♪ In that case, the spacing between buses is defined by a very different probability distribution.

The Cuernavaca bus system, the Riemann zeta function related to prime numbers, chicken retinas, and atomic nuclei are all examples of complex correlated systems.

The components of these systems aren't independent.

They interact and repel one another, and this leads to a statistical distribution in between randomness and order.

The same distribution or pattern arises even though the components of these various systems are very different.

They are said to exhibit universality.

Mathematicians model these complex correlated systems using random matrices.

The numbers in random matrices are drawn randomly from probability distributions.

The matrix might be randomly filled with zeros and ones or with any set of numbers, like the integers between one and 100.

You can characterize a matrix but its eigenvalues -- a series of numbers that can be calculated by multiplying components of the matrix together in a certain way.

Eigenvalues and random matrices are always spaced along a number line in a characteristic pattern with consecutive eigenvalues never too close together or too far apart.

The same pattern of eigenvalue spacing arises no matter how you fill the matrix with random numbers.

If you plot the distance between consecutive eigenvalues on the x-axis, and the probability of getting a particular spacing on the y-axis, the familiar lopsided curve begins to appear.

Researchers are still looking for a general answer to where this universal pattern comes from, but clues continue to emerge.

The idea of random matrix universality goes back to Eugene Wigner, a Nobel Prize-winning theoretical physicist, who worked on the Manhattan Project.

Wigner was attempting to calculate the energy levels of a uranium nucleus, which has more than 200 protons and neutrons that can arrange themselves in all different configurations.

The associated energy levels of the system were far too complex to calculate.

Wigner used random matrices instead and plotted the statistical distribution of eigenvalues.

He found that the spacing of these numbers matched the spacing of energy levels of uranium and other heavy atomic nuclei.

Two decades later, the pattern was seen in gaps between consecutive numbers called zeros of the Riemann zeta function.

These zeros are thought to control how prime numbers are distributed.

Since then, the pattern has been seen in many different settings, like in human bones and social networks.

Just recently it showed up in yet another unlikely place -- the eyes of chickens.

It was the first instance of the pattern seen in biology.

While the number line exhibits a pattern of universality in one dimension, the chicken retina cells reveals it in two dimensions.

♪♪ The color-sensitive cone cells on the chicken's retina seem haphazardly distributed, but with a remarkably uniform density.

Looking closer, the cells appear to be surrounded by what's called an exclusion region -- a space where cones with different color sensitivity can be found, but cones of the same kind cannot.

♪♪ Just how these cone cells create the exclusion zones remains a mystery, but it's similar to the repulsion between consecutive random matrix eigenvalues on the number line.

Researchers say we're just at the tip of the iceberg in understanding universality in math, physics, and even biology.