Three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors

TL;DR

This paper computes exact complex eigenvalue/vector distributions for symmetric order-three random tensors with different Lie-group invariances, using quantum field theory methods, and applies the results to determine the tensor's injective norm in the large-N limit.

Contribution

It introduces a novel method to derive closed-form eigenvalue/vector distributions for specific tensor classes using partition functions of four-fermi theories.

Findings

Exact distributions for three symmetry classes are obtained.

The injective norm of the tensor is computed in the large-N limit.

Results agree with previous numerical findings.

Abstract

Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g., tensor eigenvalue/vector distributions, are interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of zero-dimensional quantum field theories. In this paper, using the method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances, $O(N,\mathbb{R})$, $O(N,\mathbb{C})$, and $U(N,\mathbb{C})$, respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the "signed" distribution which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-$N$ limit by computing the edge of the last signed distribution, obtaining agreement with a former numerical result in the literature.