We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space.

SPhT-T05/046 ITEP/TH-??/05We present the diagrammatic technique for calculating the free energy of the Hermitian onematrix model to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves).

We compute the complete topological expansion of the formal hermitian twomatrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1 N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface. *

Bertrand EynardSPhT, CEA Saclay, France E-mail: eynard@saclay.cea.fr Abstract: We present the diagrammatic technique for calculating the free energy of the matrix eigenvalue model (the model with arbitrary power β 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).

We define a new generalized class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form x + 2 cos π/no + x −1 these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders no. In the second part of the paper, we propose the dual graph description of the corresponding Teichmüller spaces, construct the Poisson algebra of the Teichmüller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations thus providing the complete description of the above Teichmüller spaces.