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Eigenvalue distributions are important dynamical quantities in matrix models,and it is a challenging problem to derive them in tensor models. In this paper,we consider real symmetric order-three tensors with Gaussian distributions asthe simplest case, and derive an explicit formula for signed distributions ofreal tensor eigenvectors: Each real tensor eigenvector contributes to thedistribution by $\pm 1$, depending on the sign of the determinant of anassociated Hessian matrix. The formula is expressed by the confluenthypergeometric function of the second kind, which is obtained by computing apartition function of a four-fermi theory. The formula can also serve as lowerbounds of real eigenvector distributions (with no signs), and theirtightness/looseness are discussed by comparing with Monte Carlo simulations.Large-$N$ limits are taken with the characteristic oscillatory behavior of theformula being preserved.

Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory