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 a 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 drivers either speed up or wait at a stop
depending on how long ago their spy said the previous bus let.
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 any set of numbers,
like the integers between 1 and 100.
You can characterize a matrix by its eigenvalues,
a series of numbers that can be calculated
by multiplying components of the matrix
together in a certain way.
Eigenvalues of certain 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 spacings
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 zeroes 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 of a 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 can cannot.
Just how these cones 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.