How Many Eigenvalues of a Random Symmetric Tensor are Real?

TL;DR

This paper derives formulas for the expected number of real eigenvalues of random symmetric tensors and the expected absolute determinant of GOE matrices, connecting tensor eigenvalues to critical points of random polynomials.

Contribution

It provides the first closed-form formulas for the expected count of real eigenvalues of Gaussian symmetric tensors and relates this to the expected absolute determinant of GOE matrices.

Findings

Expected number of real eigenvalues for Gaussian symmetric tensors is derived.

Exact formula for the expected absolute determinant of GOE matrices is obtained.

Results connect tensor eigenvalues with critical points of random polynomials.

Abstract

This article answers a question posed by Draisma and Horobet, who asked for a closed formula for the expected number of real eigenvalues of a random real symmetric tensor drawn from the Gaussian distribution relative to the Bombieri norm. This expected value is equal to the expected number of real critical points on the unit sphere of a Kostlan polynomial. We also derive an exact formula for the expected absolute value of the determinant of a matrix from the Gaussian Orthogonal Ensemble.