Universality of the Wigner-Gurau limit for random tensors

TL;DR

This work develops a combinatorial moment method for random tensors, proving universality of the Wigner-Gurau limit: the moments of the tensor resolvent trace converge to Fuss-Catalan numbers and define a universal free Bessel law $\mu^{(p)}_\infty$. Through hypergraph and melonic/double-hypertree analysis, the authors extend Wigner’s theorem to order-$p$ tensors and establish a precise resolvent expansion with $\mathcal{R}_\infty(z)$ obeying $z^{p-2}\mathcal{R}_\infty(z)^p - z\mathcal{R}_\infty(z) + 1 = 0$. They also show a variance bound $\mathrm{Var}[\frac{1}{N} I_n(\mathcal{W}_N)]=O(1/N^2)$ and a contraction stability result: contracted tensors converge to a dilated universal law $\widetilde{\mu}_\infty$, linking the spectral behavior to matrices via $k=p-2$ contractions. The framework opens prospects for a tensor-based free probability theory, with potential central-limit, concentration, and eigenvalue questions, and establishes precise connections to melonic graph combinatorics and tensor invariants. Overall, the paper provides a robust universality principle for random tensors and lays groundwork for further spectral and probabilistic studies in high-dimensional tensor models.

Abstract

In this article, we develop a combinatorial approach for studying moments of the resolvent trace for random tensors proposed by Razvan Gurau. Our work is based on the study of hypergraphs and extends the combinatorial proof of moments convergence for Wigner's theorem. This also opens up paths for research akin to free probability for random tensors. \par Specifically, trace invariants form a complete family of tensor invariants and constitute the moments of the resolvent trace. For a random tensor with entries independent, centered, with the right variance and bounded moments, we prove the convergence of the expectation and bound the variance of the balanced single trace invariant. This is the universality of the convergence of the moments of the tensor towards the limiting moments given by the Fuss-Catalan numbers, which are the moments of the law obtained by Gurau in the Gaussian case. This generalizes Wigner's theorem for random tensors. \par Additionally, in the Gaussian case, we show that the limiting distribution of the moments of the $k$-times contracted $p$-order random tensor by a deterministic vector is always the one of a dilated Wigner-Gurau law at order $p-k$. This establishes a connection with the approach of random tensors through study of the matrix given by the $p-2$ contractions of the tensor.

Figures (4)

• Figure 1: Limit measure for $p=3$.
• Figure 2: Combinatorial maps for $p=3$ and $n=2$. The dart marks a halfedge (the root). The graph $G(b)$ associated to the two maps on the right-hand side, $2$ vertices linked by $p$ edges, is called a melon. The family of all melonic graphs is obtained recursively by inserting a melon inside the edge of a melonic.
• Figure 3: Possible partition of the edges. We have $\mathrm{Tr}_b(\mathcal{T}) = \sum_{a,b,c \atop \text{distinct}} \mathcal{T}_{aab} \mathcal{T}_{bcc} + 2 \sum_{a \neq b} \mathcal{T}_{aaa} \mathcal{T}_{abb} + \sum_{a \neq b} \mathcal{T}_{aab}^2 + \sum_{a} \mathcal{T}_{aaa}^2$.
• Figure 4: A double hypertree (above left), unfolded double hypertree (above right), melonic well partitioned (below left) and melonic graph (below right).

Theorems & Definitions (48)

• Definition 1: Symmetric tensor
• Definition 2: Trace invariant
• Theorem 1: Moments convergence
• Lemma 1
• Theorem 2: Variance
• Lemma 2
• Corollary 1: Universal measure
• Definition 3: Contraction by vectors
• Theorem 3: Limit law for the contracted tensor
• Definition 4: Multilinear transform