Random Matrices
Arizona based collaborators:
Momar Dieng
Nicholas M. Ercolani
Jason Newport
Casey Warmbrand
Survey Lectures, survey papers
Research topics
Statistical mechanics and random tilings
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.
Universality beyond the analytic class
Limit Theorems
Some of the material on this page is based on work supported in part by the National Science Foundation under grant numbers DMS-0200749, and DMS-0451495 (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.