Real eigenvalue/vector distributions of random real antisymmetric tensors

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 $e^{\mathit{NB}{h}_p\left(z_c^2/z^2\right)+o\left(N\right)}$ \( {e}^{NB\;{h}_p\left({z}_c^2/{z}^2\right)+o(N)} \) , where the function h_p(·) 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.