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MiS Preprint

76/2017

Random Spectrahedra

Paul Breiding, Khazhgali Kozhasov and Antonio Lerario

Abstract

Spectrahedral cones are linear sections of the cone of positive semidefinite symmetric matrices. We study statistical properties of random spectrahedral cones (intersected with the sphere) $$ \mathscr{S}_{\ell, n}=\{(x_0,\ldots,x_\ell)\in S^\ell\mid x_0 I + x_1 R_1 + \cdots+ x_\ell R_\ell\succ 0\}$$ where $R_1, \ldots, R_\ell$ are independent $\textrm{GOE}(n)$-distributed matrices rescaled by $(2n\ell)^{-1/2}$.

We relate the expectation of the volume of $\mathscr{S}_{\ell, n}$ with some statistics of the smallest eigenvalue of a $\textrm{GOE}(n)$ matrix, by providing explicit formulas for this quantity. These formulas imply that as $\ell,n\to \infty$ on average $\mathscr{S}_{\ell, n}$ keeps a positive fraction of the volume of the sphere $S^\ell$ (the exact constant is $\Phi(-1)\approx 0.1587$, where $\Phi$ is the cumulative distribution function of a standard gaussian variable).

For $\ell=2$ spectrahedra are generically smooth, but already when $\ell=3$ singular points on their boundaries appear with positive probability. We relate the average number $\mathbb{E} \sigma_n$ of singular points on the boundary of a three-dimensional spectrahedron $\mathscr{S}_{3,n}$ to the volume of the set of symmetric matrices whose two smallest eigenvalues coincide. In the case of quartic spectrahedra ($n=4$) we show that $\mathbb{E} \sigma_4 = 6-\frac{4}{\sqrt{3}}$. Moreover, we prove that the average number $\mathbb{E} \rho_n$ of singular points on the random symmetroid surface $$\Sigma_{3,n}=\{(x_0,x_1,x_2,x_3)\in S^3\mid \det(x_0 I + x_1 R_1 + x_2R_2 + x_3 R_3)=0\},$$ equals $n(n-1)$. This quantity is related to the volume of the set of symmetric matrices with repeated eigenvalues.