Real eigenvalue/vector distributions of random real antisymmetric tensors

2025/10/23 by Delporte, Nicolas, La Scala, Giacomo, Sasakura, Naoki +1 · 1 citation
#FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2510.20398

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_c2/z2)+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.

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• On the injective norm of random fermionic states and skew-symmetric tensors

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