MiS Preprint Repository

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.