Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory

TL;DR

This paper derives an explicit formula for the signed distribution of real eigenvectors of Gaussian symmetric tensors using a four-fermi theory, providing insights into tensor eigenvalue distributions.

Contribution

It introduces a novel explicit formula for signed eigenvector distributions of Gaussian tensors via a four-fermi theory, connecting tensor eigenvalues to hypergeometric functions.

Findings

Derived an explicit formula involving hypergeometric functions

The formula provides lower bounds for eigenvector distributions

Large-N limits preserve oscillatory behavior

Abstract

Eigenvalue distributions are important dynamical quantities in matrix models, and it is a challenging problem to derive them in tensor models. In this paper, we consider real symmetric order-three tensors with Gaussian distributions as the simplest case, and derive an explicit formula for signed distributions of real tensor eigenvectors: Each real tensor eigenvector contributes to the distribution by $\pm 1$, depending on the sign of the determinant of an associated Hessian matrix. The formula is expressed by the confluent hypergeometric function of the second kind, which is obtained by computing a partition function of a four-fermi theory. The formula can also serve as lower bounds of real eigenvector distributions (with no signs), and their tightness/looseness are discussed by comparing with Monte Carlo simulations. Large-$N$ limits are taken with the characteristic oscillatory behavior of the formula being preserved.