80 research outputs found
Non-homogenous disks in the chain of matrices
We investigate the generating functions of multi-colored discrete disks with
non-homogenous boundary conditions in the context of the Hermitian multi-matrix
model where the matrices are coupled in an open chain. We show that the study
of the spectral curve of the matrix model allows one to solve a set of loop
equations to get a recursive formula computing mixed trace correlation
functions to leading order in the large matrix limit.Comment: 25 pages, 4 figure
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
Topological expansion and boundary conditions
In this article, we compute the topological expansion of all possible
mixed-traces in a hermitian two matrix model. In other words we give a recipe
to compute the number of discrete surfaces of given genus, carrying an Ising
model, and with all possible given boundary conditions. The method is
recursive, and amounts to recursively cutting surfaces along interfaces. The
result is best represented in a diagrammatic way, and is thus rather simple to
use.Comment: latex, 25 pages. few misprints correcte
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
Large deviations of the maximal eigenvalue of random matrices
We present detailed computations of the 'at least finite' terms (three
dominant orders) of the free energy in a one-cut matrix model with a hard edge
a, in beta-ensembles, with any polynomial potential. beta is a positive number,
so not restricted to the standard values beta = 1 (hermitian matrices), beta =
1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This
model allows to study the statistic of the maximum eigenvalue of random
matrices. We compute the large deviation function to the left of the expected
maximum. We specialize our results to the gaussian beta-ensembles and check
them numerically. Our method is based on general results and procedures already
developed in the literature to solve the Pastur equations (also called "loop
equations"). It allows to compute the left tail of the analog of Tracy-Widom
laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos
corrected and preprint added ; v4 few more numbers adde
Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence
We propose an asymptotic expansion formula for matrix integrals, including
oscillatory terms (derivatives of theta-functions) to all orders. This formula
is heuristically derived from the analogy between matrix integrals, and formal
matrix models (combinatorics of discrete surfaces), after summing over filling
fractions. The whole oscillatory series can also be resummed into a single
theta function. We also remark that the coefficients of the theta derivatives,
are the same as those which appear in holomorphic anomaly equations in string
theory, i.e. they are related to degeneracies of Riemann surfaces. Moreover,
the expansion presented here, happens to be independent of the choice of a
background filling fraction.Comment: 23 pages, Late
Cut-and-Join operator representation for Kontsevich-Witten tau-function
In this short note we construct a simple cut-and-join operator representation
for Kontsevich-Witten tau-function that is the partition function of the
two-dimensional topological gravity. Our derivation is based on the Virasoro
constraints. Possible applications of the obtained expression are discussed.Comment: 5 pages, minor correction
All genus correlation functions for the hermitian 1-matrix model
We rewrite the loop equations of the hermitian matrix model, in a way which
allows to compute all the correlation functions, to all orders in the
topological expansion, as residues on an hyperelliptical curve. Those
residues, can be represented diagrammaticaly as Feynmann graphs of a cubic
interaction field theory on the curve.Comment: latex, 19 figure
Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model
In arXiv:hep-th/0310113 we started a program of creating a reference-book on
matrix-model tau-functions, the new generation of special functions, which are
going to play an important role in string theory calculations. The main focus
of that paper was on the one-matrix Hermitian model tau-functions. The present
paper is devoted to a direct counterpart for the Kontsevich and Generalized
Kontsevich Model (GKM) tau-functions. We mostly focus on calculating resolvents
(=loop operator averages) in the Kontsevich model, with a special emphasis on
its simplest (Gaussian) phase, where exists a surprising integral formula, and
the expressions for the resolvents in the genus zero and one are especially
simple (in particular, we generalize the known genus zero result to genus one).
We also discuss various features of generic phases of the Kontsevich model, in
particular, a counterpart of the unambiguous Gaussian solution in the generic
case, the solution called Dijkgraaf-Vafa (DV) solution. Further, we extend the
results to the GKM and, in particular, discuss the p-q duality in terms of
resolvents and corresponding Riemann surfaces in the example of dualities
between (2,3) and (3,2) models.Comment: 48 pages, 2 figure
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