The Normal Matrix Model, spectral statistics, and ∂- problems
The basic example for the Normal Matrix Model is the case of complex Ginibre matrices: the entries are indpendent complex Gaussians. Here are the eigenvalues of one realization of a matrix of size 200x200, and a 2D histogram of the eigenvalues after 1000 realizations.
As the matrix size grows, the eigenvalues distribute themselves
(on average) uniformly on a disc, as can be seen in this animation
of the mean density, with matrix size ranging from 1x1 to 50x50.
More general measures on the space of normal matrices can be defined. This is done by choosing a suitable function Q and defining
The induced measure on eigenvalues is then explicit:
For different choices of functions Q, the limiting density changes drastically. Here are two examples with
The analysis of these models requires the development of new
techniques for the analysis of ∂- problems in a semi-classical scaling. Strangely enough, the same semi-classical
∂- problem arises in the analysis of the integrable Davey-Stewartson II equation, as well as in other inverse problems.
Some of the material on this page is based on work supported
in part by the National Science Foundation under grant numbers
DMS-0200749, DMS-0451495, DMS-0800979, and DMS-1401268 (McLaughlin). Any opinions,
findings, and conclusions or recommendations expressed in this material
are those of the authors and do not necessarily reflect the views of
the National Science Foundation.
Statistical mechanics and random tilings
Given side lengths a, b, and c, how many tilings are there? (Answer is called the MacMahon formula.)
Is there any characteristic behavior that is true of most tilings, when the mesh size shrinks to zero?
One may tile a hexagonal domain with rhombi of three orientations if the hexagon has opposing sides of equal length, with all side lengths being integers.
To visualize this, one may fill the hexagon with equilateral triangles (with unit side length).
Then one forms rhombi by gluing together pairs of triangles.
Thinking like a probabilist, one asks:
Notice that in each corner there is an abundance of rhombi of the same orientation. While it is a little hard to see in the middle image above, once the mesh is small, the frozen regions are quite easy to see.
The existence of frozen or "arctic" regions has been established (see the work of Cohn, Larsen, and Propp
) and more recently this has been extended to other planar domains as well. Fluctuations of the random boundaries of these arctic zones have been studied by Johansson
and by Baik, Kriecherbauer, McLaughlin, and Miller
Combinatorics and 2D quantum gravity
In the remarkable work of
Bessis, Itzykson, and Zuber,
the connection between matrix integrals and graphical enumeration was explained, and a great many formulae were set down. This pioneering work has set the stage for a huge number of developments, both in the physics literature as well as in the rigorous mathematical analysis of random matrices. The fundamental combinatorial problem is to enumerate maps (graphs embedded into Riemann surfaces) according to vertex valences and genus of the underlying Riemann surface. From the physics literature there arose the claim (viewed as a conjecture in the mathematical community) that the partition function of random matrix theory possesses an asymptotic expansion in even powers of 1/N, whose coefficients are generating functions for these graphical enumeration problems. This was put on a rigorous mathematical footing by Ercolani and McLaughlin, using Riemann-Hilbert techniques.