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Double Scaling in Tensor Models with a Quartic Interaction

Stephane Dartois, Razvan Gurau, Vincent Rivasseau

TL;DR

This paper analyzes the $1/N$ expansion of random tensor models with quartic interactions and identifies the subleading cherry-tree graphs that dominate the double scaling limit for dimensions $D<6$. Using the Loop Vertex Expansion and constructive QFT tools, the authors reorganize the perturbative series via pruning and reduction to reduced graphs, isolating cherry trees as the leading non-melonic contributions and deriving their explicit amplitudes. They prove precise bounds showing non-cherry graphs are subleading by at least $N^{-1/2}$, and compute the cherry-tree contributions to obtain a closed double scaling limit with a square-root singularity around a critical point $x_c$, mirroring the melonic behavior but with new multi-scaling possibilities. The results indicate a stable double scaling in tensor models up to $D<6$, with the upper critical dimension at $D=6$, and pave the way for further multi-scaling analyses and potential new phases in higher-dimensional random geometries.

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.

Double Scaling in Tensor Models with a Quartic Interaction

TL;DR

This paper analyzes the expansion of random tensor models with quartic interactions and identifies the subleading cherry-tree graphs that dominate the double scaling limit for dimensions . Using the Loop Vertex Expansion and constructive QFT tools, the authors reorganize the perturbative series via pruning and reduction to reduced graphs, isolating cherry trees as the leading non-melonic contributions and deriving their explicit amplitudes. They prove precise bounds showing non-cherry graphs are subleading by at least , and compute the cherry-tree contributions to obtain a closed double scaling limit with a square-root singularity around a critical point , mirroring the melonic behavior but with new multi-scaling possibilities. The results indicate a stable double scaling in tensor models up to , with the upper critical dimension at , and pave the way for further multi-scaling analyses and potential new phases in higher-dimensional random geometries.

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.

Paper Structure

This paper contains 13 sections, 7 theorems, 81 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Matrix Models
  3. Tensor Models
  4. Constructive QFT
  5. Graph classification and the main result
  6. Pruning and Grafting
  7. Reduction
  8. Cherry Trees
  9. Reduced graphs amplitudes
  10. Proof of Theorem \ref{['maintheo']}
  11. Bounds
  12. Computation of Cherry Trees
  13. The Double Scaling Limit

Key Result

Lemma 1

where $w^T_{\ell}(\{u\})$ is the infimum over the $u_{\ell'}$ parameters over the lines $\ell'$ forming the unique path $P^T_\ell$ in $T$ joining the ends of $\ell$.

Figures (11)

  • Figure 1: The Quartic Melonic Interactions
  • Figure 2: A LVE tree in the direct (left) and intermediate field (right) representation for a quartic interaction.
  • Figure 3: Pruning
  • Figure 4: Grafting
  • Figure 5: Reduction of a pruned graph
  • ...and 6 more figures

Theorems & Definitions (12)

  • Lemma 1: RZTW
  • Theorem 1: The Mixed Expansion, Gurau
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • Definition 4
  • Lemma 3
  • Theorem 2
  • Definition 5
  • ...and 2 more