Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps

Ordinary tensor models of rank D >= 3 are dominated at large N by tree-like graphs, known as melonic triangulations. We here show that non-melonic contributions can be enhanced consistently, leading to different types of large N limits. We first study the most generic quartic model at D = 4, with maximally enhanced non-melonic interactions. The existence of the 1/N expansion is proved and we further characterize the dominant triangulations. This combinatorial analysis is then used to define a non-quartic, non-melonic class of models for which the large N free energy and the relevant expectations can be calculated explicitly. They are matched with random matrix models which contain multi-trace invariants in their potentials: they possess a branched polymer phase and a 2D quantum gravity phase, and a transition between them whose entropy exponent is positive. Finally, a non-perturbative analysis of the generic quartic model is performed, which proves analyticity in the coupling constants in cardioid domains. (C) 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license