Real tensor eigenvalue/vector distributions of the Gaussian tensor model via a four-fermi theory

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.