Joint distributions of eigenvectors of symmetric random tensors

TL;DR

This paper computes the joint distributions of eigenvectors of symmetric random tensors using quantum field theory, revealing universal behaviors and providing asymptotic formulas validated by simulations.

Contribution

It extends previous work on mean distributions to joint distributions, deriving universal formulas and asymptotics for eigenvector distributions of symmetric tensors.

Findings

Derived joint eigenvector distributions for symmetric tensors

Established universality extending from mean to joint distributions

Validated results with Monte Carlo simulations

Abstract

We compute the joint distributions of arbitrary numbers of eigenvectors of real and complex symmetric random tensors by the quantum field theoretical methods which 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 by Monte Carlo computations.