Real eigenvalue/vector distributions of random real antisymmetric tensors

TL;DR

This paper analyzes the distribution of real eigenvalues and eigenvectors of Gaussian random real antisymmetric tensors using quantum field theory, revealing universal asymptotic behaviors and applications to tensor norms.

Contribution

It provides analytic expressions for eigenvalue distributions of antisymmetric tensors and uncovers universality in large-$N$ asymptotics across tensor types.

Findings

Derived finite-$N$ signed eigenvalue distribution formulas.

Established large-$N$ asymptotic forms for eigenvalue distributions.

Identified universality in eigenvalue distribution behavior across tensor classes.

Abstract

Real eigenpairs of a real antisymmetric tensor of order $p$ and dimension $N$ can be defined as pairs of a real eigenvalue and $p$ orthonormal $N$-dimensional real eigenvectors. We compute the signed and the genuine distributions of such eigenvalues of Gaussian random real antisymmetric tensors by using a quantum field theoretical method. An analytic expression for finite $N$ is obtained for the signed distribution and the analytic large-$N$ asymptotic forms for both. We compute the edge of the distribution for large-$N$, one application of which is to give an upper bound (believed tight) of the injective norm of the random real antisymmetric tensor. We find a large-$N$ universality across various tensor eigenvalue distributions: the large-$N$ asymptotic forms of the distributions of the eigenvalues $z$ of the complex, complex symmetric, real symmetric, and real antisymmetric random tensors are all expressed by $e^{N\,B\, h_p(z_c^2/z^2)+o(N)}$, where the function $h_p(\cdot)$ depends only on the order $p$, while $B$ and $z_c$ differ for each case, $NB$ being the total dimension of the eigenvectors and $z_c$ being determined by the phase transition point of the quantum field theory.