2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals

We prove that the 2-hermitean matrix model and the complex-matrix model obey the same loop equations, and as a byproduct, we find a formula for Itzykzon-Zuber's type integrals over the unitary group. Integrals over U(n) are rewritten as gaussian integrals over triangular matrices and then computed explicitly. That formula is an efficient alternative to the former Shatashvili's formula.Comment: 29 pages, Late

Correlation Functions of Complex Matrix Models

For a restricted class of potentials (harmonic+Gaussian potentials), we express the resolvent integral for the correlation functions of simple traces of powers of complex matrices of size $N$, in term of a determinant; this determinant is function of four kernels constructed from the orthogonal polynomials corresponding to the potential and from their Cauchy transform. The correlation functions are a sum of expressions attached to a set of fully packed oriented loops configurations; for rotational invariant systems, explicit expressions can be written for each configuration and more specifically for the Gaussian potential, we obtain the large $N$ expansion ('t Hooft expansion) and the so-called BMN limit.Comment: latex BMN.tex, 7 files, 6 figures, 30 pages (v2 for spelling mistake and added reference) [http://www-spht.cea.fr/articles/T05/174

A matrix model for the topological string II: The spectral curve and mirror geometry

In a previous paper, we presented a matrix model reproducing the topological string partition function on an arbitrary given toric Calabi-Yau manifold. Here, we study the spectral curve of our matrix model and thus derive, upon imposing certain minimality assumptions on the spectral curve, the large volume limit of the BKMP "remodeling the B-model" conjecture, the claim that Gromov-Witten invariants of any toric Calabi-Yau 3-fold coincide with the spectral invariants of its mirror curve.Comment: 1+37 page

Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups

The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials prod tr{X^{p_1} Omega Y^{q_1} Omega^dagger X^{p_2} ... with the weight exp tr{X Omega Y Omega^dagger} are computed for the orthogonal and symplectic groups. We proceed in two steps. First, the integral over the compact group is recast into a Gaussian integral over strictly upper triangular complex matrices (with some additional symmetries), supplemented by a summation over the Weyl group. This result follows from the study of loop equations in an associated two-matrix integral and may be viewed as the adequate version of Duistermaat-Heckman's theorem for our correlation function integrals. Secondly, the Gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions.Comment: 58 pages; Acknowledgements added; small corrections in appendix A; minor changes & Note Adde

A Matrix model for plane partitions

We construct a matrix model equivalent (exactly, not asymptotically), to the random plane partition model, with almost arbitrary boundary conditions. Equivalently, it is also a random matrix model for a TASEP-like process with arbitrary boundary conditions. Using the known solution of matrix models, this method allows to find the large size asymptotic expansion of plane partitions, to ALL orders. It also allows to describe several universal regimes.Comment: Latex, 41 figures. Misprints and corrections. Changing the term TASEP to self avoiding particle porces

Topological expansion of the chain of matrices

We solve the loop equations to all orders in 1/N 2, for the Chain of Matrices matrix model (with possibly an external field coupled to the last matrix of the chain). We show that the topological expansion of the free energy, is, like for the 1 and 2-matrix model, given by the symplectic invariants of [19]. As a consequence, we find the double scaling limit explicitly, and we discuss modular properties, large N asymptotics. We also briefly discuss the limit of an infinite chain of matrices (matrix quantum mechanics)

Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups

58 pagesThe Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials prod tr{X^{p_1} Omega Y^{q_1} Omega^dagger X^{p_2} ... with the weight exp tr{X Omega Y Omega^dagger} are computed for the orthogonal and symplectic groups. We proceed in two steps. First, the integral over the compact group is recast into a Gaussian integral over strictly upper triangular complex matrices (with some additional symmetries), supplemented by a summation over the Weyl group. This result follows from the study of loop equations in an associated two-matrix integral and may be viewed as the adequate version of Duistermaat-Heckman's theorem for our correlation function integrals. Secondly, the Gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions