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arXiv:2605.09804 · Progress of Theoretical and Experimental Physics

Joint Distributions of Eigenvectors of Symmetric Random Tensors

Naoki Sasakura

Published Jul 27, 2026• 0 citations
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Abstract

Abstract We compute the joint distributions of arbitrary numbers of eigenvectors of real and complex symmetric random tensors by the quantum field theoretical methods that were previously used to compute the mean distributions. We obtain the random matrix representations and the large-dimension asymptotics of the joint distributions. The latter can be expressed by a common function of tensor geometries, extending the universality found for the mean distributions to the joint distributions. Several crosschecks of our results are carried out using Monte Carlo computations.

DOI
10.1093/ptep/ptag143

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Abstract We compute the joint distributions of arbitrary numbers of eigenvectors of real and complex symmetric random tensors by the quantum field theoretical methods that were previously used to compute the mean distributions. We obtain the random matrix representations and the large-dimension asymptotics of the joint distributions. The latter can be expressed by a common function of tensor geometries, extending the universality found for the mean distributions to the joint distributions. Several crosschecks of our results are carried out using Monte Carlo computations.
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We obtain exact analytic expressions of real tensor eigenvalue/vector distributions of real symmetric order-three tensors with Gaussian distributions for N ≤ 8. This is achieved by explicitly computing the partition function of a zero-dimensional boson–fermion system with four interactions. The distributions are expressed by combinations of polynomial, exponential, and error functions as results of feasible complicated bosonic integrals that appear after fermionic integrations. By extrapolating the expressions and also using a previous result, we guess a large-N expression. The expressions are compared with Monte Carlo simulations, and precise agreement and good agreement are obtained with the exact and the large-N expressions, respectively. Understanding the feasibility of the integration is left for future study, which would provide a general-N analytic formula.
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We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study. Finally, we present some applications to the theory of random quantum states in quantum information theory.