The 1/N Expansion of Tensor Models Beyond Perturbation Theory

TL;DR

This paper rigorously analyzes the 1/N expansion of tensor models beyond perturbation theory, introducing a mixed expansion method that simplifies the calculation of subleading corrections and establishes universality in the large N limit.

Contribution

It introduces a new mixed expansion technique for tensor models, enabling explicit series in 1/N and proving Borel summability and universality beyond perturbation theory.

Findings

Cumulants expressed as explicit 1/N series with bounded remainders

Perturbative expansion shown to be Borel summable

Tensor models fall into Gaussian universality class at large N

Abstract

We analyze in full mathematical rigor the most general quartically perturbed invariant probability measure for a random tensor. Using a version of the Loop Vertex Expansion (which we call the mixed expansion) we show that the cumulants write as explicit series in 1/N plus bounded rest terms. The mixed expansion recasts the problem of determining the subleading corrections in 1/N into a simple combinatorial problem of counting trees decorated by a finite number of loop edges. As an aside, we use the mixed expansion to show that the (divergent) perturbative expansion of the tensor models is Borel summable and to prove that the cumulants respect an uniform scaling bound. In particular the quartically perturbed measures fall, in the N to infinity limit, in the universality class of Gaussian tensor models.