Tensor models with generalized melonic interactions

TL;DR

This paper introduces generalized melonic interactions in tensor models, characterizes their Feynman graphs, and proves Gaussian large N limits for a subclass, extending the universality class and connecting to matrix models.

Contribution

It defines and characterizes generalized melonic interactions, proves Gaussian limits for totally unbalanced cases, and links tensor models to simplified matrix models.

Findings

Large N limit is Gaussian for totally unbalanced interactions.

Feynman graphs are tree-like and bijective with trees in the Gaussian case.

Tensor models can be reformulated as matrix models with fewer degrees of freedom.

Abstract

Tensor models are natural generalizations of matrix models. The interactions and observables in the case of unitary invariant models are generalizations of matrix traces. Some notable interactions in the literature include the melonic ones, the tetrahedral one as well as the planar ones in rank three, or necklaces in even ranks. Here we introduce generalized melonic interactions which generalize the melonic and necklace interactions. We characterize them as tree-like gluings of quartic interactions. We also completely characterize the Feynman graphs which contribute to the large $N$ limit. For a subclass of generalized melonic interactions called totally unbalanced interactions, we prove that the large $N$ limit is Gaussian and therefore the Feynman graphs are in bijection with trees. This result further extends the class of tensor models which fall into the Gaussian universality class. Another key aspect of tensor models with generalized melonic interactions is that they can be written as matrix models without increasing the number of degrees of freedom of the original tensor models. In the case of totally unbalanced interactions, this new matrix model formulation in fact decreases the number of degrees of freedom, meaning that some of the original degrees of freedom are effectively integrated. We then show how the large $N$ Gaussian behavior can be reproduced using a saddle point analysis on those matrix models.