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

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

Large N asymptotics of orthogonal polynomials, from integrability to algebraic geometry

In this short lecture, we compute asymptotics of orthogonal polynomials, from a saddle point approximation. This is an example of a calculation which shows the link between integrability, algebraic geometry and random matrices.Comment: Proceedings Les Houches sumer school, Applications of Random Matrices in Physics, June 6-25 200

Hamiltonian Cycles on a Random Three-coordinate Lattice

Consider a random three-coordinate lattice of spherical topology having 2v vertices and being densely covered by a single closed, self-avoiding walk, i.e. being equipped with a Hamiltonian cycle. We determine the number of such objects as a function of v. Furthermore we express the partition function of the corresponding statistical model as an elliptic integral.Comment: 10 pages, LaTeX, 3 eps-figures, one reference adde

Large N expansion of the 2-matrix model, multicut case

We present a method, based on loop equations, to compute recursively, all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model, in the case where the support of the density of eigenvalues is not connected. We illustrate the method by computing the free energy of a statistical physics model on a discretized torus.Comment: latex, 1 figur

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

Recursion between Mumford volumes of moduli spaces

We propose a new proof, as well as a generalization of Mirzakhani's recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich's integral, i.e. we relate them to a Ribbon graph decomposition of Riemann surfaces. We find a generalization of Mirzakhani's recursions to measures containing all higher Mumford's kappa classes, and not only kappa1 as in the Weil-Petersson case.Comment: Latex, 18 page