  Quantum field theories can be applied to compute various statistical
properties of random tensors. In particular signed distributions of tensor
eigenvalues/vectors are the easiest, which can be computed as partition
functions of four-fermi theories. Though signed distributions are different
from genuine ones because of extra signs of weights, they are expected to
coincide in vicinities of ends of distributions. In this paper, we perform a
case study of the signed eigenvalue/vector distribution of the real symmetric
order-three random tensor. The correct critical point and the correct end in
the large $N$ limit are obtained from the four-fermi theory, for which a method
using the Schwinger-Dyson equation is very efficient. Since locations of ends
are particularly important in applications, such as the largest eigenvalues and
the best rank-one tensor approximations, signed distributions are the easiest
and highly useful through the Schwinger-Dyson method.
