  We prove the existence of a 1/N expansion to all orders in beta matrix models
with a confining, off-critical potential corresponding to an equilibrium
measure with a connected support. Thus, the coefficients of the expansion can
be obtained recursively by the "topological recursion" of Chekhov and Eynard.
Our method relies on the combination of a priori bounds on the correlators and
the study of Schwinger-Dyson equations, thanks to the uses of classical complex
analysis techniques. These a priori bounds can be derived following Boutet de
Monvel, Pastur and Shcherbina, or for strictly convex potentials by using
concentration of measure. Doing so, we extend the strategy of Guionnet and
Maurel-Segala, from the hermitian models (beta = 2) and perturbative
potentials, to general beta models. The existence of the first correction in
1/N has been considered previously by Johansson and more recently by
Kriecherbauer and Shcherbina. Here, by taking similar hypotheses, we extend the
result to all orders in 1/N.
