Orthogonal Polynomial Method and Odd Vertices in Matrix Models

The method of orthogonal polynomials is a powerful technique for the non perturbative integration of matrix models over one [1] or more matrices [2] in particular with even potential, i.e. with vertices with an even number of legs. Indeed, with even potential, the calculation simplifies both because the integrals are well defined and, as we shall see, the number of equations needed to solve the problem is smaller. On the other hand the model with odd vertices, in particular with cubic vertices is more natural in a number of problems; e.g. in the dynamical triangulation model of quantum gravity, where the random surface is given by a polyhedron with triangular faces, the order of the vertices appearing in the dual graphs is always three. Brézin et al. [3] solved the problem with cubic vertices using the saddle point technique. Bessis [4] introduced an alternative method (the orthogonal polynomial method) which to some extent appears more powerful e.g. in dealing with matrix model with more than one matrix variable [2]. In particular, the orthogonal polynomial method has been proved useful in the treatment of a cubic vertex two-matrix model [5] in the context of the Ising model on a random planar lattice. The purpose of this paper is to show, in a systematic way, how to extend the orthogonal polynomial method to arbitrary vertices, both even and odd and any combination of them. We shall follow the article of Bessis et al. [1] generalizing some aspects to the case of odd vertices, in particular we shall recover, for the simplest case of cubic vertices, the result of [3] for spherical topology. Hopefully such a treatment can be extended to higher genus. The use of mixed vertices e.g. cubic plus quartic vertex, allows us to write a well defined i.e. convergent, partition function by adding to the cubic interaction a quartic term which makes the action bounded from