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.
