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Random tensor models in the large N limit: Uncoloring the colored tensor models

Valentin Bonzom, Razvan Gurau, Vincent Rivasseau

TL;DR

This work generalizes the 1/N expansion from colored tensor models to invariant tensor models with a single generic complex tensor, showing that melonic graphs dominate at large N and produce a universal continuum limit with γ_{melons} = 1/2. It derives a self-consistent dressed two-point function U, establishes Virasoro constraints for the large-N free energy, and demonstrates a family of multicritical points with γ_m = 1 - 1/m, mirroring matrix-model universality in higher dimensions. Colors are shown to be a bookkeeping device rather than a fundamental feature, enabling a robust geometrogenesis framework in arbitrary dimensions. The results lay the groundwork for constructive analyses and connect tensor models to branched-polymer universality through multicritical behavior, enriching the landscape of random geometry beyond two dimensions.

Abstract

Tensor models generalize random matrix models in yielding a theory of dynamical triangulations in arbitrary dimensions. Colored tensor models have been shown to admit a 1/N expansion and a continuum limit accessible analytically. In this paper we prove that these results extend to the most general tensor model for a single generic, i.e. non-symmetric, complex tensor. Colors appear in this setting as a canonical book-keeping device and not as a fundamental feature. In the large N limit, we exhibit a set of Virasoro constraints satisfied by the free energy and an infinite family of multicritical behaviors with entropy exponents γ_m=1-1/m.

Random tensor models in the large N limit: Uncoloring the colored tensor models

TL;DR

This work generalizes the 1/N expansion from colored tensor models to invariant tensor models with a single generic complex tensor, showing that melonic graphs dominate at large N and produce a universal continuum limit with γ_{melons} = 1/2. It derives a self-consistent dressed two-point function U, establishes Virasoro constraints for the large-N free energy, and demonstrates a family of multicritical points with γ_m = 1 - 1/m, mirroring matrix-model universality in higher dimensions. Colors are shown to be a bookkeeping device rather than a fundamental feature, enabling a robust geometrogenesis framework in arbitrary dimensions. The results lay the groundwork for constructive analyses and connect tensor models to branched-polymer universality through multicritical behavior, enriching the landscape of random geometry beyond two dimensions.

Abstract

Tensor models generalize random matrix models in yielding a theory of dynamical triangulations in arbitrary dimensions. Colored tensor models have been shown to admit a 1/N expansion and a continuum limit accessible analytically. In this paper we prove that these results extend to the most general tensor model for a single generic, i.e. non-symmetric, complex tensor. Colors appear in this setting as a canonical book-keeping device and not as a fundamental feature. In the large N limit, we exhibit a set of Virasoro constraints satisfied by the free energy and an infinite family of multicritical behaviors with entropy exponents γ_m=1-1/m.

Paper Structure

This paper contains 14 sections, 5 theorems, 46 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The $1/N$ Expansion of Invariant Tensor Models
  3. Tensor invariants and action
  4. Graph amplitudes
  5. Topology from bubbles
  6. Large N limit
  7. The melonic family and the large $N$ factorization
  8. Combinatorial description of melons
  9. Large $N$ factorization
  10. The leading order two-point function and free energy
  11. The continuum limit
  12. Virasoro constraints
  13. Multicritical behaviors in the large N limit
  14. Combinatorics of colored graphs: jackets, degree and melons

Key Result

Proposition 1

Let ${\cal B}$ a $D$-colored graph with $2p$ vertices. Then the total number of faces of ${\cal B}$ respects

Figures (7)

  • Figure 1: Graphical representation of trace invariants.
  • Figure 2: A Feynman graph.
  • Figure 3: Trace invariants and gluings of simplices in $D=3$.
  • Figure 4: Stranded graph and a non-bipartite invariant.
  • Figure 5:
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5