Double Scaling in Tensor Models with a Quartic Interaction

TL;DR

This paper explores the detailed structure of subleading contributions in the 1/N expansion of quartic tensor models, revealing a stable double scaling limit and uncovering new multi-scaling phenomena.

Contribution

It identifies and analyzes subleading graphs called cherry trees in tensor models, establishing a stable double scaling limit and highlighting an upper critical dimension of 6.

Findings

Double scaling limit is stable unlike in matrix models.

Cherry trees govern subleading behavior for D<6.

Singularity at fixed distance from the origin indicates complex multi-scaling phenomena.

Abstract

In this paper we identify and analyze in detail the subleading contributions in the 1/N expansion of random tensors, in the simple case of a quartically interacting model. The leading order for this 1/N expansion is made of graphs, called melons, which are dual to particular triangulations of the D-dimensional sphere, closely related to the "stacked" triangulations. For D<6 the subleading behavior is governed by a larger family of graphs, hereafter called cherry trees, which are also dual to the D-dimensional sphere. They can be resummed explicitly through a double scaling limit. In sharp contrast with random matrix models, this double scaling limit is stable. Apart from its unexpected upper critical dimension 6, it displays a singularity at fixed distance from the origin and is clearly the first step in a richer set of yet to be discovered multi-scaling limits.