Depending on attendance we will either meet in McLaughlin's office or in a conference room to be determined.

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 , and

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.
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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:

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.

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.