Exact analytic expressions of real tensor eigenvalue distributions of Gaussian tensor model for small $N$

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.