513 research outputs found
Loop equations for the semiclassical 2-matrix model with hard edges
The 2-matrix models can be defined in a setting more general than polynomial
potentials, namely, the semiclassical matrix model. In this case, the
potentials are such that their derivatives are rational functions, and the
integration paths for eigenvalues are arbitrary homology classes of paths for
which the integral is convergent. This choice includes in particular the case
where the integration path has fixed endpoints, called hard edges. The hard
edges induce boundary contributions in the loop equations. The purpose of this
article is to give the loop equations in that semicassical setting.Comment: Latex, 20 page
Mixed correlation functions in the 2-matrix model, and the Bethe ansatz
Using loop equation technics, we compute all mixed traces correlation
functions of the 2-matrix model to large N leading order. The solution turns
out to be a sort of Bethe Ansatz, i.e. all correlation functions can be
decomposed on products of 2-point functions. We also find that, when the
correlation functions are written collectively as a matrix, the loop equations
are equivalent to commutation relations.Comment: 38 pages, LaTex, 24 figures. misprints corrected, references added, a
technical part moved to appendi
Mixed Correlation Functions of the Two-Matrix Model
We compute the correlation functions mixing the powers of two non-commuting
random matrices within the same trace. The angular part of the integration was
partially known in the literature: we pursue the calculation and carry out the
eigenvalue integration reducing the problem to the construction of the
associated biorthogonal polynomials. The generating function of these
correlations becomes then a determinant involving the recursion coefficients of
the biorthogonal polynomials.Comment: 16 page
Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case
In this article, we show that the double scaling limit correlation functions
of a random matrix model when two cuts merge with degeneracy (i.e. when
for arbitrary values of the integer ) are the same as the
determinantal formulae defined by conformal models. Our approach
follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and
uses a Lax pair representation of the conformal models (giving
Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in
\cite{BleherEynard}. In particular we define Baker-Akhiezer functions
associated to the Lax pair to construct a kernel which is then used to compute
determinantal formulae giving the correlation functions of the double scaling
limit of a matrix model near the merging of two cuts.Comment: 37 pages, 4 figures. Presentation improved, typos corrected.
Published in Journal Of Statistical Mechanic
Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
We compute the generating functions of a O(n) model (loop gas model) on a
random lattice of any topology. On the disc and the cylinder, they were already
known, and here we compute all the other topologies. We find that the
generating functions (and the correlation functions of the lattice) obey the
topological recursion, as usual in matrix models, i.e they are given by the
symplectic invariants of their spectral curve.Comment: pdflatex, 89 pages, 12 labelled figures (15 figures at all), minor
correction
Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
We solve the loop equations of the -ensemble model analogously to the
solution found for the Hermitian matrices . For \beta=1y^2=U(x)\beta((\hbar\partial)^2-U(x))\psi(x)=0\hbar\propto
(\sqrt\beta-1/\sqrt\beta)/Ny^2-U(x)[y,x]=\hbarF_h-expansion at arbitrary . The set of "flat"
coordinates comprises the potential times and the occupation numbers
\widetilde{\epsilon}_\alpha\mathcal F_0\widetilde{\epsilon}_\alpha$.Comment: 58 pages, 7 figure
Matrix eigenvalue model: Feynman graph technique for all genera
We present the diagrammatic technique for calculating the free energy of the
matrix eigenvalue model (the model with arbitrary power by the
Vandermonde determinant) to all orders of 1/N expansion in the case where the
limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint
intervals (curves).Comment: Latex, 27 page
Holomorphic anomaly and matrix models
The genus g free energies of matrix models can be promoted to modular
invariant, non-holomorphic amplitudes which only depend on the geometry of the
classical spectral curve. We show that these non-holomorphic amplitudes satisfy
the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We
derive as well holomorphic anomaly equations for the open string sector. These
results provide evidence at all genera for the Dijkgraaf--Vafa conjecture
relating matrix models to type B topological strings on certain local
Calabi--Yau threefolds.Comment: 23 pages, LaTex, 3 figure
Duality, Biorthogonal Polynomials and Multi-Matrix Models
The statistical distribution of eigenvalues of pairs of coupled random
matrices can be expressed in terms of integral kernels having a generalized
Christoffel--Darboux form constructed from sequences of biorthogonal
polynomials. For measures involving exponentials of a pair of polynomials V_1,
V_2 in two different variables, these kernels may be expressed in terms of
finite dimensional ``windows'' spanned by finite subsequences having length
equal to the degree of one or the other of the polynomials V_1, V_2. The
vectors formed by such subsequences satisfy "dual pairs" of first order systems
of linear differential equations with polynomial coefficients, having rank
equal to one of the degrees of V_1 or V_2 and degree equal to the other. They
also satisfy recursion relations connecting the consecutive windows, and
deformation equations, determining how they change under variations in the
coefficients of the polynomials V_1 and V_2. Viewed as overdetermined systems
of linear difference-differential-deformation equations, these are shown to be
compatible, and hence to admit simultaneous fundamental systems of solutions.
The main result is the demonstration of a spectral duality property; namely,
that the spectral curves defined by the characteristic equations of the pair of
matrices defining the dual differential systems are equal upon interchange of
eigenvalue and polynomial parameters.Comment: Latex, 44 pages, 1 tabl
Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem
We consider biorthogonal polynomials that arise in the study of a
generalization of two--matrix Hermitian models with two polynomial potentials
V_1(x), V_2(y) of any degree, with arbitrary complex coefficients. Finite
consecutive subsequences of biorthogonal polynomials (`windows'), of lengths
equal to the degrees of the potentials, satisfy systems of ODE's with
polynomial coefficients as well as PDE's (deformation equations) with respect
to the coefficients of the potentials and recursion relations connecting
consecutive windows. A compatible sequence of fundamental systems of solutions
is constructed for these equations. The (Stokes) sectorial asymptotics of these
fundamental systems are derived through saddle-point integration and the
Riemann-Hilbert problem characterizing the differential equations is deduced.Comment: v1:41 pages, 5 figures, 1 table. v2:Typos and other errors corrected.
v3: Some conceptual changes, added appendix and two figures v4: Minor
typographical changes, improved figures. v5: updated version (submitted) 49
pages, 7 figures, 1 tabl
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