Random Matrices in 2D, Laplacian Growth and Operator Theory

Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.Comment: 88 pages, 8 figure

A new transform approach to biharmonic boundary value problems in circular domains with applications to Stokes flows

In this thesis, we present a new transform approach for solving biharmonic boundary value problems in two-dimensional polygonal and circular domains. Our approach provides a unified general approach to finding quasi-analytical solutions to a wide range of problems in Stokes flows and plane elasticity. We have chosen to analyze various Stokes flow problems in different geometries which have been solved using other techniques and present our transform approach to solve them. Our approach adapts mathematical ideas underlying the Unified transform method, also known as the Fokas method, due to Fokas and collaborators in recent years. We first consider Stokes flow problems in polygonal domains whose boundaries consist of straight line edges. We show how to solve problems in the half-plane subject to different boundary conditions along the real axis and we are able to retrieve analytical results found using other techniques. Next, we present our transform approach to solve for a flow past a periodic array of semi-infinite plates and for a periodic array of point singularities in a channel, followed by a brief discussion on how to systematically solve problems in more complex channel geometries. Next, we show how to solve problems in circular domains whose boundaries consist of a combination of straight line and circular edges. We analyze the problems of a flow past a semicircular ridge in the half-plane, a translating and rotating cylinder above a wall and a translating and rotating cylinder in a channel.Open Acces

Selection of the ground state for nonlinear Schroedinger equations

We prove for a class of nonlinear Schr\"odinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as {\it ground state selection}. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear Master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree-Fock type.Comment: Revision of 2001 preprint; 108 pages Te

Carleman factorization of layer potentials on smooth domains

One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincar\'e operator into a product of two self-adjoint transforms. Resurrecting some pertinent indications of Carleman and M. G. Krein, we exploit this grossly overlooked structure by confining the spectral analysis of the Neumann-Poincar\'e operator to the amenable $L^2$-space setting, rather than bouncing back and forth the computations between Sobolev spaces of negative or positive fractional order. An enhanced, fresh new look at symmetrizable linear transforms enters into the picture in the company of geometric-microlocal analysis techniques. The outcome is manyfold, complementing recent advances on the theory of layer potentials, in the smooth boundary setting

Orthogonal polynomials, equilibrium measures and quadrature domains associated with random matrix models

Motivated by asymptotic questions related to the spectral theory of complex random matrices, this work focuses on the asymptotic analysis of orthogonal polynomials with respect to quasi-harmonic potentials in the complex plane. The ultimate goal is to develop new techniques to obtain strong asymptotics (asymptotic expansions valid uniformly on compact subsets) for planar orthogonal polynomials and use these results to understand the limiting behavior of spectral statistics of matrix models as their size goes to infinity. For orthogonal polynomials on the real line the powerful Riemann-Hilbert approach is the main analytic tool to derive asymptotics for the eigenvalue correlations in Hermitian matrix models. As yet, no such method is available to obtain asymptotic information about planar orthogonal polynomials, but some steps in this direction have been taken. The results of this thesis concern the connection between the asymptotic behavior of orthogonal polynomials and the corresponding equilibrium measure. It is conjectured that this connection is established via a quadrature identity: under certain conditions the weak-star limit of the normalized zero counting measure of the orthogonal polynomials is a quadrature measure for the support of the equilibrium measure of the corresponding two-dimensional electrostatic variational problem of the underlying potential. Several results are presented on equilibrium measures, quadrature domains, orthogonal polynomials and their relation to matrix models. In particular, complete strong asymptotics are obtained for the simplest nontrivial quasi-harmonic potential by a contour integral reduction method and the Riemann-Hilbert approach, which confirms the above conjecture for this special cas

Differentiable positive definite kernels on two-point homogeneous spaces

In this work we study continuous kernels on compact two-point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the d-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order ⌊(d−1)/2⌋ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two-point homogeneous spaces.CNPq (grant 141908/2015-7)FAPESP (grant 2014/00277-5

A PDE Approach to the Combinatorics of the Full Map Enumeration Problem: Exact Solutions and their Universal Character

Maps are polygonal cellular networks on Riemann surfaces. This paper completes a program of constructing closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the tiling polygons. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable {\it unwinding identity} in terms of the Appell polynomials generated by Bessel functions. Our treatment, based on random matrix theory and Riemann-Hilbert problems for orthogonal polynomials reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis as well as directly related to universality results for random matrix spectra as described by Deift, Kriecherbauer, McLaughlin, Venakides and Zhou