7 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
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
On the Algebraic--Geometrical Solutions of the sine--Gordon Equation
We examine the relation between two known classes of solutions of the
sine--Gordon equation, which are expressed by theta functions on hyperelliptic
Riemann surfaces. The first one is a consequence of the Fay's trisecant
identity. The second class exists only for odd genus hyperelliptic Riemann
surfaces which admit a fixed--point--free automorphism of order two. We show
that these two classes of solutions coincide. The hyperelliptic surfaces
corresponding to the second class appear to be double unramified coverings of
the Riemann surfaces corresponding to the first class of solutions. We also
discuss the soliton limits of these solutions.Comment: 10 pages, LaTex, SISSA--ISAS 10/94/EP (Revised version: the reference
[12] added; small changes in the Introduction
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