Random Matrices

Arizona based collaborators:

Joe Gibney
Nick Ercolani

Research topics

  • The Normal Matrix Model, spectral statistics, and - problems

    Normal Matrix Model N=200

    Normal Matrix Model N=200
    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.

    NMM Density animation
    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

    Meausre on matrices

    The induced measure on eigenvalues is then explicit:

    NMM EV measure
    For different choices of functions Q, the limiting density changes drastically. Here are two examples with Quartic Q, and Quadratic Q:

    NMM Density animation

    NMM Density animation

    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.

  • Statistical mechanics and random tilings

    triangular lattice within hexagon
    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).
    tiled hexagonThen one forms rhombi by gluing together pairs of triangles.
    Thinking like a probabilist, one asks:

  • 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?
  • tiled hexagon 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.

    Research links

    Recent Publications on RMT

  • Asymptotic analysis of random matrices with external source and a family of algebraic curves
  • A quick derivation of the loop equations for random matrices (with N. M. Ercolani).
  • Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices (with N. M. Ercolani and V. U. Pierce).
  • Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles (with J. Baik, T. Kriecherbauer, and P. Miller).
  • Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results(with J. Baik, T. Kriecherbauer, and P. Miller).
  • Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration(with N. M. Ercolani).