  We obtain exact analytic expressions of real tensor eigenvalue/vector
distributions of real symmetric order-three tensors with Gaussian distributions
for $N\leq 8$. This is achieved by explicitly computing the partition function
of a zero-dimensional boson-fermion system with four-interactions. The
distributions are expressed by combinations of polynomial, exponential and
error functions as results of feasible complicated bosonic integrals which
appear after fermionic integrations. By extrapolating the expressions and also
using a previous result, we guess a large-$N$ expression. The expressions are
compared with Monte Carlo simulations, and precise and good agreement are
obtained with the exact and the large-$N$ expressions, respectively.
Understanding the feasibility of the integration is left for future study,
which would provide a general-$N$ analytic formula.

High Energy Physics - Theory