536 research outputs found

### 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

### 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

### Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

We solve the loop equations of the hermitian 2-matrix model to all orders in
the topological $1/N^2$ expansion, i.e. we obtain all non-mixed correlation
functions, in terms of residues on an algebraic curve. We give two
representations of those residues as Feynman-like graphs, one of them involving
only cubic vertices.Comment: 48 pages, LaTex, 68 figure

### 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

### Intersection numbers of spectral curves

We compute the symplectic invariants of an arbitrary spectral curve with only
1 branchpoint in terms of integrals of characteristic classes in the moduli
space of curves. Our formula associates to any spectral curve, a characteristic
class, which is determined by the laplace transform of the spectral curve. This
is a hint to the key role of Laplace transform in mirror symmetry. When the
spectral curve is y=\sqrt{x}, the formula gives Kontsevich--Witten intersection
numbers, when the spectral curve is chosen to be the Lambert function
\exp{x}=y\exp{-y}, the formula gives the ELSV formula for Hurwitz numbers, and
when one chooses the mirror of C^3 with framing f, i.e.
\exp{-x}=\exp{-yf}(1-\exp{-y}), the formula gives the Marino-Vafa formula, i.e.
the generating function of Gromov-Witten invariants of C^3. In some sense this
formula generalizes ELSV, Marino-Vafa formula, and Mumford formula.Comment: 53 pages, 1 fig, Latex, minor modification

### 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

### 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 $\beta$ 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

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