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

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 $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinantal formulae defined by conformal $(2m,1)$ 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 $(2m,1)$ 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

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

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 $1/N^2$ 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

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

The Potts-q random matrix model : loop equations, critical exponents, and rational case

In this article, we study the q-state Potts random matrix models extended to branched polymers, by the equations of motion method. We obtain a set of loop equations valid for any arbitrary value of q. We show that, for q=2-2 \cos {l \over r} \pi (l, r mutually prime integers with l < r), the resolvent satisfies an algebraic equation of degree 2 r -1 if l+r is odd and r-1 if l+r is even. This generalizes the presently-known cases of q=1, 2, 3. We then derive for any 0 \leq q \leq 4 the Potts-q critical exponents and string susceptibility.Comment: 7 pages, submitted to Phys. Letters