8 research outputs found
Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy
We continue the analysis of the spectral curve of the normal random matrix
ensemble, introduced in an earlier paper. Evolution of the full quantum curve
is given in terms of compatibility equations of independent flows. The
semiclassical limit of these flows is expressed through canonical differential
forms of the spectral curve. We also prove that the semiclassical limit of the
evolution equations is equivalent to Whitham hierarchy.Comment: 14 page
Generic critical points of normal matrix ensembles
The evolution of the degenerate complex curve associated with the ensemble at
a generic critical point is related to the finite time singularities of
Laplacian Growth. It is shown that the scaling behavior at a critical point of
singular geometry is described by the first Painlev\'e
transcendent. The regularization of the curve resulting from discretization is
discussed.Comment: Based on a talk given at the conference on Random Matrices, Random
Processes and Integrable Systems, CRM Montreal, June 200
Normal random matrix ensemble as a growth problem
In general or normal random matrix ensembles, the support of eigenvalues of
large size matrices is a planar domain (or several domains) with a sharp
boundary. This domain evolves under a change of parameters of the potential and
of the size of matrices. The boundary of the support of eigenvalues is a real
section of a complex curve. Algebro-geometrical properties of this curve encode
physical properties of random matrix ensembles. This curve can be treated as a
limit of a spectral curve which is canonically defined for models of finite
matrices. We interpret the evolution of the eigenvalue distribution as a growth
problem, and describe the growth in terms of evolution of the spectral curve.
We discuss algebro-geometrical properties of the spectral curve and describe
the wave functions (normalized characteristic polynomials) in terms of
differentials on the curve. General formulae and emergence of the spectral
curve are illustrated by three meaningful examples.Comment: 44 pages, 14 figures; contains the first part of the original file.
The second part will be submitted separatel
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