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Universality for Random Tensors

Razvan Gurau

TL;DR

The paper extends universality from random matrices to random tensors of any rank $D$ by developing a $1/N$ expansion in the tensor setting and introducing a comprehensive graphical formalism using $D+1$-colored graphs. It proves two universality theorems: (i) i.i.d. tensor entries with covariance $\sigma^2$ converge to a Gaussian tensor model, and (ii) joint invariant distributions with uniformly bounded cumulants converge to Gaussian limits with covariance $K(\mathcal{B}^{(2)})$, highlighting that covariance is distribution-dependent and not universal. Central to the approach is the melonic (degree-zero) structure and the notion of minimal covering graphs, which govern leading large-$N$ behavior, with boundary (open) graphs and jackets providing the combinatorial backbone. These results establish a robust Gaussian universality framework for random tensors, enabling controlled analysis of high-dimensional random geometries and their relevance to statistical mechanics and quantum gravity, while clarifying how finite-$N$ details shape the limiting covariance.

Abstract

We prove two universality results for random tensors of arbitrary rank D. We first prove that a random tensor whose entries are N^D independent, identically distributed, complex random variables converges in distribution in the large N limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large N limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large N Gaussian is not universal, but depends strongly on the details of the joint distribution.

Universality for Random Tensors

TL;DR

The paper extends universality from random matrices to random tensors of any rank by developing a expansion in the tensor setting and introducing a comprehensive graphical formalism using -colored graphs. It proves two universality theorems: (i) i.i.d. tensor entries with covariance converge to a Gaussian tensor model, and (ii) joint invariant distributions with uniformly bounded cumulants converge to Gaussian limits with covariance , highlighting that covariance is distribution-dependent and not universal. Central to the approach is the melonic (degree-zero) structure and the notion of minimal covering graphs, which govern leading large- behavior, with boundary (open) graphs and jackets providing the combinatorial backbone. These results establish a robust Gaussian universality framework for random tensors, enabling controlled analysis of high-dimensional random geometries and their relevance to statistical mechanics and quantum gravity, while clarifying how finite- details shape the limiting covariance.

Abstract

We prove two universality results for random tensors of arbitrary rank D. We first prove that a random tensor whose entries are N^D independent, identically distributed, complex random variables converges in distribution in the large N limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large N limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large N Gaussian is not universal, but depends strongly on the details of the joint distribution.

Paper Structure

This paper contains 27 sections, 18 theorems, 161 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Notation and Main Theorems
  3. Random Matrices
  4. Universality for Random Matrices
  5. Random Tensors
  6. Closed $D+1$-colored Graphs
  7. Melons
  8. Open graphs and the boundary graph
  9. Gaussian Distribution for Tensors
  10. Melonic observables
  11. Arbitrary observables
  12. Proof of Theorem \ref{['thm:mic']}
  13. Melonic observables: direct computation
  14. Arbitrary observables
  15. Proof of Theorem \ref{['thm:mare']}
  16. ...and 12 more sections

Key Result

Theorem 1

Consider $N^D$ i.i.d. random variables $T_{\vec{n} }$, each of covariance $\sigma^2$. Then, in the large $N$ limit, the tensor $\mathbb{T}_{\vec{n}} = \frac{1}{ N^{ \frac{D-1 }{2} } } T_{\vec{n}}$ converges in distribution to a Gaussian tensor of covariance $\sigma^2$.

Figures (20)

  • Figure 1: Graphical representation of trace invariants.
  • Figure 2: A covering graph ${\cal G }$ of an observable ${\cal B}$.
  • Figure 3: Doubled graphs contributing to an observable.
  • Figure 4: The graphs ${\cal G }$ and ${\cal J }$, ${\cal G }^{(q)'}$ and ${\cal J }_{ {\cal G }^{(q)'} }$ and ${\cal G }^{(q)}$ and ${\cal J }_{ {\cal G }^{(q)} }$.
  • Figure 5: The graph ${\cal G }^{(q) }$ and the graphs ${\cal G }^{(q,q')}$.
  • ...and 15 more figures

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1: Universality 1
  • Theorem 2: Main Theorem: Universality 2
  • Definition 6
  • Theorem 3
  • Definition 7
  • ...and 19 more