Multiple scaling limits of $U(N{)}^2\times O(D)$ multi-matrix models

Abstract

We study the double- and triple-scaling limits of a complex multi-matrix model, with $\mathrm{U}(N{)}^2\times \mathrm{O}(D)$ symmetry. The double-scaling limit amounts to taking simultaneously the large-$N$ (matrix size) and large-$D$ (number of matrices) limits while keeping the ratio $N/\sqrt{D}=M$ fixed. The triple-scaling limit consists in taking the large-$M$ limit while tuning the coupling constant $\lambda$ to its critical value ${\lambda }_c$ and keeping fixed the product $M({\lambda }_c-\lambda {)}^{\alpha }$, for some value of $\alpha$ that depends on the particular combinatorial restrictions imposed on the model. Our first main result is the complete recursive characterization of the Feynman graphs of arbitrary genus which survive in the double-scaling limit. Next, we classify all the dominant graphs in the triple-scaling limit, which we find to have a plane binary tree structure with decorations. Their critical behavior belongs to the universality class of branched polymers. Lastly, we classify all the dominant graphs in the triple-scaling limit under the restriction to three-edge connected (or two-particle irreducible) graphs. Their critical behavior falls in the universality class of Liouville quantum gravity (or, in other words, the Brownian sphere).