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

Abstract

Abstract Eigenvalue distributions are important dynamic 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 a 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.