$O(N)$ Random Tensor Models

TL;DR

This paper introduces a new class of $O(N)$ invariant tensor models with a large $N$ expansion, analyzing their critical behavior and identifying leading and next-to-leading order graphs using combinatorics techniques.

Contribution

It defines a novel class of tensor models with $O(N)^{ imes 3}$ invariance, establishes their large $N$ expansion, and analyzes their critical properties at leading and NLO.

Findings

Existence of a large $N$ expansion for the model.

Identification of leading and NLO graphs.

Computation of critical exponents at both orders.

Abstract

We define in this paper a class of three indices tensor models, endowed with $O(N)^{\otimes 3}$ invariance ($N$ being the size of the tensor). This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model (and hence of the colored model) and the $U(N)$ invariant models. We first exhibit the existence of a large $N$ expansion for such a model with general interactions. We then focus on the quartic model and we identify the leading and next-to-leading order (NLO) graphs of the large $N$ expansion. Finally, we prove the existence of a critical regime and we compute the critical exponents, both at leading order and at NLO. This is achieved through the use of various analytic combinatorics techniques.