Real eigenvalue/vector distributions of random real antisymmetric tensors

TL;DR

This work defines real eigenpairs of a real antisymmetric tensor as a real eigenvalue together with $p$ orthonormal eigenvectors and solves for their distributions using a quantum-field-theoretic framework. It derives finite-$N$ expressions for the signed distribution and large-$N$ asymptotics for both signed and genuine distributions, including an edge analysis that bounds the injective norm of the random tensor. A key outcome is a universality: at large $N$ the eigenvalue distributions across real/complex and symmetric/antisymmetric tensor ensembles share a common exponential form $\rho(\nu) \sim e^{N B h_p(x)}$, with a model-dependent $B$ and edge point $x$ dictated by a phase transition. The results connect tensor eigenvalue problems to quantum-field methods, Schwinger-Dyson analysis, and potential applications to quantum information through geometric entanglement measures of random fermionic multipartite states.

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.

Figures (2)

• Figure 1: Comparison of the analytic expression \ref{['eq:analSignedRho']} (in continuous blue) of the signed eigenvalue distributions for $N=4,5,6,7$, with the numerical simulation (in dotted red, with associated numerical errors, done for 10000 iterations)
• Figure 2: Comparison of the analytic expression \ref{['eq:analSignedRho']} (in continuous blue) of the signed eigenvalue distributions for $N=4,5,6,7$, and $p=4$ with the numerical simulation (in dotted red, with associated numerical errors, done for 10000 iterations)