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Eigenvalue distributions are important dynamical quantities in matrix models,and it is an interesting challenge to study corresponding quantities in tensormodels. We study real tensor eigenvalue/vector distributions for real symmetricorder-three random tensors with the Gaussian distribution as the simplest case.We first rewrite this problem as the computation of a partition function of afour-fermi theory with $R$ replicated fermions. The partition function isexactly computed for some small-$N,R$ cases, and is shown to precisely agreewith Monte Carlo simulations. For large-$N$, it seems difficult to compute itexactly, and we apply an approximation using a self-consistency equation fortwo-point functions and obtain an analytic expression. It turns out that thereal tensor eigenvalue distribution obtained by taking $R=1/2$ is simply theGaussian within this approximation. We compare the approximate expression withMonte Carlo simulations, and find that, if an extra overall factor depending on$N$ is multiplied to the the expression, it agrees well with the Monte Carloresults. It is left for future study to improve the approximation for large-$N$to correctly derive the overall factor.

Real tensor eigenvalue/vector distributions of the Gaussian tensor model via a four-fermi theory