Regular colored graphs of positive degree

Abstract

Regular colored graphs are dual representations of pure colored $D$-dimensional complexes. These graphs can be classified with respect to a positive integer, their degree, much like maps are characterized by the genus. We analyze the structure of regular colored graphs of fixed degree and perform their exact and asymptotic enumeration. In particular we show that the generating function of the family of graphs of fixed degree is an algebraic series with a positive radius of convergence, independent of the degree. We describe the singular behavior of this series near its dominant singularity, and use the results to establish the double scaling limit of colored tensor models: interestingly the behavior is qualitatively very different for $3\le D\le 5$ and for $D\ge 6$.