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

TL;DR

This paper investigates the distribution of eigenvalues and eigenvectors in Gaussian random tensors using a four-fermi theory approach, providing exact results for small cases and an analytic approximation for large tensors, validated by simulations.

Contribution

It introduces a novel four-fermi theory framework to analyze real tensor eigenvalue distributions, offering exact solutions for small tensors and an analytic approximation for large tensors, with validation against Monte Carlo simulations.

Findings

Exact partition function computed for small-$N,R$ cases.

Analytic expression approximated for large-$N$ tensors.

Approximate distribution matches Monte Carlo results with a scaling factor.

Abstract

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.