Real eigenvalue/vector distributions of random real antisymmetric tensors

A bstract 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 $$ {e}^{NB\;{h}_p\left({z}_c^2/{z}^2\right)+o(N)} $$ e NB h p z c 2 / z 2 + 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.