  Eigenvalue distributions are important dynamical quantities in matrix models,
and it is an interesting challenge to study corresponding quantities in tensor
models. We study real tensor eigenvalue/vector distributions for real symmetric
order-three random tensors with the Gaussian distribution as the simplest case.
We first rewrite this problem as the computation of a partition function of a
four-fermi theory with $R$ replicated fermions. The partition function is
exactly computed for some small-$N,R$ cases, and is shown to precisely agree
with Monte Carlo simulations. For large-$N$, it seems difficult to compute it
exactly, and we apply an approximation using a self-consistency equation for
two-point functions and obtain an analytic expression. It turns out that the
real tensor eigenvalue distribution obtained by taking $R=1/2$ is simply the
Gaussian within this approximation. We compare the approximate expression with
Monte 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 Carlo
results. It is left for future study to improve the approximation for large-$N$
to correctly derive the overall factor.
