This year, the University of Chicago saw an odd coincidence: Two similarly
named mathematical scientists, Persi Diaconis and Percy Deift, came to give
distinguished lecture series--one in statistics, the other in mathematics.
They each talked about random matrix theory and its relation to other topics
in physics and mathematics, including quantum gravity, the Riemann zeta
function, and integrable systems.
Some explanations are in order. This zeta function is one of the central
elements of pure mathematics. It is defined in terms of prime numbers,
and it has an infinite number of zeros arrayed along a line in the complex
plane. There are formulas that enable one to translate statistical information
about the spacing of zeros into knowledge about the spacing of prime numbers,
and vice versa.
The random matrix idea grew out of Eugene Wigner's investigations of the highly
excited states of nuclei. A nucleus is too complex for us to even give an
accurate representation of its Hamiltonian. In 1951, Wigner hypothesized that
different nuclei might behave as if the Hamiltonian of each one was a random
matrix, with each matrix having been picked from an ensemble determined by the
Hamiltonian's symmetry properties. Freeman Dyson noticed that there are three
such ensembles that make especial physical and mathematical sense. The
statistical distributions of energy levels generated by these ensembles show
highly structured spectra with a considerable tendency for neighboring levels
to repel one another and spread out as much as they can. The statistical properties
of any stretch of a few tens or hundreds of consecutive highly excited levels
tends to be universal, that is, independent of everything but the symmetry of the
Hamiltonian and the average level spacing. The
same statistical distribution could be used to describe the spacing of zeros of
the zeta function. Andrey Nickolayevich Kolmogorov proposed a roughly similar
kind of universality for the fine structure of turbulence problems. Since these
great works, universality has been sort of expected in hydrodynamics and
statistical physics problems.
Peter Sarnak and Zeev Rudnick constructed theorems that support Dyson's ideas
about the relation between the distributions from the zeta function and from the
matrices. Diaconis, in one of his Chicago lectures, described how he tested the
hypothesis of the identity of these distributions using some elegant applications
of modern statistical analysis. These numerical tests, which followed up on earlier
work by Hugh Montgomery and Michael Berry, show the robustness and power of the
connection proposed by Dyson.
But there is more to this universality. Not only is the Wigner/Dyson answer
applicable to almost all matrices, but it applies to many other problems as well,
including quantum chaos and universal conductance fluctuations. Moreover, random
matrices have eigenvalues whose distribution is given by a statistical mechanics
problem involving particles with Coulomb-like interactions. These problems fit
into the general class of exactly solvable statistical mechanics models, including,
for example, the two-dimensional Ising model and two-dimensional quantum gravity.
These models can, in turn, be converted into problems in Hamiltonian dynamics by a
trick equivalent to Feynman's path-integral formulation of quantum mechanics.
More yet: In his lectures, Deift pointed out that the random matrix problem has a
natural connection to problems of integrable systems. He discussed "the remarkable
discovery of Gardner, Greene, Kruskal and Miura in 1967 of a way to integrate the
KortewegdeVries equation, a particular infinite dimensional Hamiltonian system
arising in the theory of water waves." Amazingly, this work gives a method for
finding the complete spacetime solution of this nonlinear partial differential
equation. In hindsight, we see that the exact solutions can also be found by
constructing all the different integrals of the motion, or by developing a
description of the independent modes of excitation of the system (as in the seminal
1965 work of Martin Kruskal and Norman Zabusky on solitons), or by several other
apparently unconnected methods. Many different dynamical problems have been solved
by these methods. The solution of these problems is one of the great achievements
of the mathematical and physical sciences in the second half of the 20th
century. According to Deift, all these integrable systems fit naturally into the same
skein of ideas as problems in random matrix theory, statistical physics, quantum gravity,
function theory, and many more areas.
Edouard Brézin and Anthony Zee have used renormalization arguments to understand the
random matrix and Hamiltonian universalities. They describe these problems by indicating
their relation to the exactly solvable "Gaussian model" of statistical mechanics. They
explain universality as a consequence of the robustness of Gaussian probability
distributions under changes of variables. Even though we do understand many aspects
of this universality, one can marvel that so many different problem areas are all
interconnected in this wonderful and apparently magic fashion. This "magic" signals that
there are probably additional deep connections, yet to be discovered.