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The rank three tensor model with tetrahedral interaction was shown byCarrozza and Tanasa to admit a $1/N$ expansion, dominated by melonic diagrams,and double tadpoles decorated with melons at next-to-leading order. This modelhas generated a renewed interest in tensor models because it has the same large$N$ limit as the SYK model. In contrast with matrix models, there is no methodwhich would be able to prove the existence of $1/N$ expansions in arbitrarytensor models. The method used by Carrozza and Tanasa proves the existence ofthe $1/N$ expansion using two-dimensional topology, before identifying theleading order and next-to-leading graphs. However, another method was requiredfor complex, rank three tensor models with planar interactions, which is basedon flips. The latter are moves which cut two propagators of Feynman graphs andreglue them differently. They allow transforming graphs while tracking theirorders in the $1/N$ expansion. Here we use this method to re-prove the resultsof Carrozza and Tanasa, thereby proving the existence of the $1/N$ expansion,the melonic dominance at leading order and the melon-decorated double tadpolesat next-to-leading order, all in one go.

Another proof of the $1/N$ expansion of the rank three tensor model with tetrahedral interaction