On the Gaussian surface area of spectrahedra

TL;DR

This paper investigates the Gaussian surface area of spectrahedra, demonstrating that for large dimensions, the surface area of a random spectrahedron with Gaussian orthogonal ensemble matrices scales as a specific power of the dimension with high probability.

Contribution

It establishes a probabilistic bound on the Gaussian surface area of spectrahedra in high dimensions, revealing its asymptotic behavior for large matrix sizes.

Findings

Gaussian surface area scales as Θ(n^{1/8}) for large n

Spectrahedra with Gaussian orthogonal ensemble matrices exhibit predictable surface area behavior

Results hold with high probability for sufficiently large dimensions

Abstract

We show that for sufficiently large $n\geq 1$ and $d=C n^{3/4}$ for some universal constant $C>0$, a random spectrahedron with matrices drawn from Gaussian orthogonal ensemble has Gaussian surface area $\Theta(n^{1/8})$ with high probability.