Universality of the Wigner-Gurau limit for random tensors

TL;DR

This paper proves the universality of the Wigner-Gurau limit for random tensors, extending Wigner's theorem using combinatorial hypergraph methods and establishing convergence of tensor moments to Fuss-Catalan distributions.

Contribution

It introduces a combinatorial approach for tensor moments, proving their convergence to Fuss-Catalan moments and generalizing Wigner's theorem for random tensors.

Findings

Convergence of tensor moments to Fuss-Catalan distribution.

Universality of the Wigner-Gurau limit for random tensors.

Connection between tensor contractions and limiting distributions.

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. \\ 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. \\ 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 matrices given by the $p-2$ contractions of the tensor.