Isosystolic inequalities and the topological expansion for random surface and matrix models

Using the isosystolic inequalities on Riemann surfaces, we prove that for many random surface or matrix models the radius of convergence of the perturbative series at fixed genus is independent of the genus. This result applies for instance to the dynamically triangulated random surface model in any dimension or to many matrix models with regular propagators in the superrenormalizable domain, for instance ³ in dimensiond<6, in dimensiond<4, and various otherP()₂ models (in particular all those containing an odd power of ). We hope that this result is a first step towards a more rigorous understanding of the genus dependence of surface models or of quantum gravity coupled with matter fields.