Real eigenvalue/vector distributions of random real antisymmetric tensors

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.