1,214 research outputs found
The PDEs of biorthogonal polynomials arising in the two-matrix model
The two-matrix model can be solved by introducing bi-orthogonal polynomials.
In the case the potentials in the measure are polynomials, finite sequences of
bi-orthogonal polynomials (called
"windows") satisfy polynomial ODEs as well as deformation equations (PDEs)
and finite difference equations (Delta-E) which are all Frobenius compatible
and define discrete and continuous isomonodromic deformations for the irregular
ODE, as shown in previous works of ours.
In the one matrix model an explicit and concise expression for the
coefficients of these systems is known and it allows to relate the partition
function with the isomonodromic tau-function of the overdetermined system.
Here, we provide the generalization of those expressions to the case of
bi-orthogonal polynomials, which enables us to compute the determinant of the
fundamental solution of the overdetermined system of ODE+PDEs+Delta-E.Comment: 20 pages v1 18 Nov 2003; v2 9 Jan 2004: trivial Latex mistake
correcte
The Cauchy two-matrix model
We introduce a new class of two(multi)-matrix models of positive Hermitean
matrices coupled in a chain; the coupling is related to the Cauchy kernel and
differs from the exponential coupling more commonly used in similar models. The
correlation functions are expressed entirely in terms of certain biorthogonal
polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving
the way to a steepest descent analysis and universality results. The
interpretation of the formal expansion of the partition function in terms of
multicolored ribbon-graphs is provided and a connection to the O(1) model. A
steepest descent analysis of the partition function reveals that the model is
related to a trigonal curve (three-sheeted covering of the plane) much in the
same way as the Hermitean matrix model is related to a hyperelliptic curve.Comment: 34 pages, 2 figures. V2: changes only to metadat
Large N expansion of the 2-matrix model
We present a method, based on loop equations, to compute recursively all the
terms in the large topological expansion of the free energy for the
2-hermitian matrix model. We illustrate the method by computing the first
subleading term, i.e. the free energy of a statistical physics model on a
discretized torus.Comment: 41 pages, 9 figures eps
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
Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes
Random Hermitian matrices with a source term arise, for instance, in the
study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and
sample covariance matrices \cite{Baik:2005}.
We consider the case when the external source matrix has two
distinct real eigenvalues: with multiplicity and zero with multiplicity
. The source is small in the sense that is finite or , for . For a Gaussian potential, P\'ech\'e
\cite{Peche:2006} showed that for sufficiently small (the subcritical
regime) the external source has no leading-order effect on the eigenvalues,
while for sufficiently large (the supercritical regime) eigenvalues
exit the bulk of the spectrum and behave as the eigenvalues of
Gaussian unitary ensemble (GUE). We establish the universality of these results
for a general class of analytic potentials in the supercritical and subcritical
regimes.Comment: 41 pages, 4 figure
Moment determinants as isomonodromic tau functions
We consider a wide class of determinants whose entries are moments of the
so-called semiclassical functionals and we show that they are tau functions for
an appropriate isomonodromic family which depends on the parameters of the
symbols for the functionals. This shows that the vanishing of the tau-function
for those systems is the obstruction to the solvability of a Riemann-Hilbert
problem associated to certain classes of (multiple) orthogonal polynomials. The
determinants include Haenkel, Toeplitz and shifted-Toeplitz determinants as
well as determinants of bimoment functionals and the determinants arising in
the study of multiple orthogonality. Some of these determinants appear also as
partition functions of random matrix models, including an instance of a
two-matrix model.Comment: 24 page
The dependence on the monodromy data of the isomonodromic tau function
[Note: important Corrigendum now available at arXiv:1601.04790] The
isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the
Malgrange's divisor of generalized monodromy data for which a vector bundle is
nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no
solution. In their original work, Jimbo, Miwa, Ueno did not derive the
dependence on the (generalized) monodromy data (i.e. monodromy representation
and Stokes' parameters). We fill the gap by providing a (simpler and more
general) description in which all the parameters of the problem
(monodromy-changing and monodromy-preserving) are dealt with at the same level.
We thus provide variational formulae for the isomonodromic tau function with
respect to the (generalized) monodromy data. The construction applies more
generally: given any (sufficiently well-behaved) family of Riemann-Hilbert
problems (RHP) where the jump matrices depend arbitrarily on deformation
parameters, we can construct a one-form Omega (not necessarily closed) on the
deformation space (Malgrange's differential), defined off Malgrange's divisor.
We then introduce the notion of discrete Schlesinger transformation: it means
that we allow the solution of the RHP to have poles (or zeros) at prescribed
point(s). Even if Omega is not closed, its difference evaluated along the
original solution and the transformed one, is shown to be the logarithmic
differential (on the deformation space) of a function. As a function of the
position of the points of the Schlesinger transformation, yields a natural
generalization of Sato formula for the Baker-Akhiezer vector even in the
absence of a tau function, and it realizes the solution of the RHP as such BA
vector. Some exemples (Painleve' II and finite Toplitz/Hankel determinants) are
provided.Comment: 34 pages, 7 figures. An important "Corrigendum" is now available as
arXiv:1601.0479
Meromorphic differentials with imaginary periods on degenerating hyperelliptic curves
We provide a direct and explicit proof that imaginary (real) normalized differentials of the second kind with prescribed polar part do not develop additional singularities as the underlying hyperelliptic Riemann surface degenerates in an arbitrary way
Mixed correlation function and spectral curve for the 2-matrix model
We compute the mixed correlation function in a way which involves only the
orthogonal polynomials with degrees close to , (in some sense like the
Christoffel Darboux theorem for non-mixed correlation functions). We also
derive new representations for the differential systems satisfied by the
biorthogonal polynomials, and we find new formulae for the spectral curve. In
particular we prove the conjecture of M. Bertola, claiming that the spectral
curve is the same curve which appears in the loop equations.Comment: latex, 1 figure, 55 page
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