18 research outputs found
On Associativity Equations in Dispersionless Integrable Hierarchies
We discuss the origin of the associativity (WDVV) equations in the context of
quasiclassical or Whitham hierarchies. The associativity equations are shown to
be encoded in the dispersionless limit of the Hirota equations for KP and Toda
hierarchies. We show, therefore, that any tau-function of dispersionless KP or
Toda hierarchy provides a solution to associativity equations. In general, they
depend on infinitely many variables. We also discuss the particular solution to
the dispersionless Toda hierarchy that describes conformal mappings and
construct a family of new solutions to the WDVV equations depending on finite
number of variables.Comment: 16 pages, LaTe
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
Large scale correlations in normal and general non-Hermitian matrix ensembles
We compute the large scale (macroscopic) correlations in ensembles of normal
random matrices with an arbitrary measure and in ensembles of general
non-Hermition matrices with a class of non-Gaussian measures. In both cases the
eigenvalues are complex and in the large limit they occupy a domain in the
complex plane. For the case when the support of eigenvalues is a connected
compact domain, we compute two-, three- and four-point connected correlation
functions in the first non-vanishing order in 1/N in a manner that the
algorithm of computing higher correlations becomes clear. The correlation
functions are expressed through the solution of the Dirichlet boundary problem
in the domain complementary to the support of eigenvalues. The two-point
correlation functions are shown to be universal in the sense that they depend
only on the support of eigenvalues and are expressed through the Dirichlet
Green function of its complement.Comment: 16 pages, 1 figure, LaTeX, submitted to J. Phys. A special issue on
random matrices, minor corrections, references adde
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
Integrability in SFT and new representation of KP tau-function
We are investigating the properties of vacuum and boundary states in the CFT
of free bosons under the conformal transformation. We show that transformed
vacuum (boundary state) is given in terms of tau-functions of dispersionless KP
(Toda) hierarchies. Applications of this approach to string field theory is
considered. We recognize in Neumann coefficients the matrix of second
derivatives of tau-function of dispersionless KP and identify surface states
with the conformally transformed vacuum of free field theory.Comment: 25 pp, LaTeX, reference added in the Section 3.
Second and Third Order Observables of the Two-Matrix Model
In this paper we complement our recent result on the explicit formula for the
planar limit of the free energy of the two-matrix model by computing the second
and third order observables of the model in terms of canonical structures of
the underlying genus g spectral curve. In particular we provide explicit
formulas for any three-loop correlator of the model. Some explicit examples are
worked out.Comment: 22 pages, v2 with added references and minor correction