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Center for Nonlinear Studies |
The limiting behaviour of the spectral correlation functions of unitarily invariant Hermitian random matrix models as the size of the matrices tends to infinity is determined by the asymptotic properties of the associated orthogonal polynomials in a suitable scaling limit. The global asymptotic behaviour of these polynomials is governed by a measure on the real line which is the unique solution of an ambient two-dimensional electrostatic equilibrium problem in the complex plane, restricted to the real axis. For more general random normal matrix models with unitarily invariant measures, the orthogonality measure for the associated orthogonal polyomials is supported on the whole complex plane. The relevant equilibrium measure in this case is still the well-studied one for log-Coulomb gases in a background potential, but this provides only partial information: we still lack a general method for determining the asymptotics of the planar orthogonal polynomials.
For a special class of background potentials the equilibrium measure is uniform (i.e. constant) on a compact set whose boundary is an analytic curve. This particular case is connected to the so-called Laplacian Growth.
Our primary goal is to understand the asymptotic properties of the orthogonal polynomials in the global scaling limit. In particular, there is a very interesting, yet rigorously unproven, localization phenomenon: the zeroes of the orthogonal polynomials are accumulating along some critical trajectories encoded into the geometry of the support of the equilibrium measure and the so-called wave function probability densities are converging to the (singular) conformal measure of the same domain.
We present strong numerical evidence supporting these conjectures and show some applications relevant to shape recognition and image
reconstruction and to Laplacian Growth.
Host: Razvan Teodorescu, CNLS and T-13