  Random tensor models have applications in a variety of fields, such as
quantum gravity, quantum information theory, mathematics of modern
technologies, etc., and studying their statistical properties, e.g., tensor
eigenvalue/vector distributions, are interesting and useful. Recently some
tensor eigenvalue/vector distributions have been computed by expressing them as
partition functions of zero-dimensional quantum field theories. In this paper,
using the method, we compute three cases of complex eigenvalue/vector
distributions of symmetric order-three random tensors, where the three cases
can be characterized by the Lie-group invariances, $O(N,\mathbb{R})$,
$O(N,\mathbb{C})$, and $U(N,\mathbb{C})$, respectively. Exact closed-form
expressions of the distributions are obtained by computing partition functions
of four-fermi theories, where the last case is of the "signed" distribution
which counts the distribution with a sign factor coming from a Hessian matrix.
As an application, we compute the injective norm of the complex symmetric
order-three random tensor in the large-$N$ limit by computing the edge of the
last signed distribution, obtaining agreement with a former numerical result in
the literature.
