Double scaling limit of multi-matrix models at large D

Abstract

In this paper, we study a double scaling limit of two multi-matrix models: the -invariant model with all quartic interactions and the bipartite -invariant model with tetrahedral interaction (D being here the number of matrices and N being the size of each matrix). Those models admit a double, large N and large D expansion. While N tracks the genus of the Feynman graphs, D tracks another quantity called the grade. In both models, we rewrite the sum over Feynman graphs at fixed genus and grade as a finite sum over combinatorial objects called schemes. This is a result of combinatorial nature which remains true in the quantum mechanical setting and in quantum field theory. Then we proceed to the double scaling limit at large D, i.e. for vanishing grade. In particular, we find that the most singular schemes, in both models, are the same as those found in Benedetti et al for the -invariant model restricted to its tetrahedral interaction. This is a different universality class than in the 1-matrix model whose double scaling is not summable.