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) $${\mathcal{S}}_{\ell ,n}=\{(x_0,\dots ,x_{\ell })\in S^{\ell }\mid x_{0}I+x_1{R}_1+\cdots +x_{\ell }{R}_{\ell }\succ 0\}$$ where $R_1,\dots ,R_{\ell }$ are independent $\text{GOE}(n)$-distributed matrices rescaled by $(2n\ell {)}^{-1/2}$.

We relate the expectation of the volume of ${\mathcal{S}}_{\ell ,n}$ with some statistics of the smallest eigenvalue of a $\text{GOE}(n)$ matrix, by providing explicit formulas for this quantity. These formulas imply that as $\ell ,n\to \mathrm{\infty }$ on average ${\mathcal{S}}_{\ell ,n}$ keeps a positive fraction of the volume of the sphere $S^{\ell }$ (the exact constant is $\mathrm{\Phi }(-1)\approx 0.1587$, where $\mathrm{\Phi }$ 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 ${\mathcal{S}}_{3,n}$ 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 $${\mathrm{\Sigma }}_{3,n}=\{(x_0,x_1,x_2,x_3)\in S^3\mid det(x_{0}I+x_1{R}_1+x_2{R}_2+x_3{R}_3)=0\},$$ equals $n(n-1)$. This quantity is related to the volume of the set of symmetric matrices with repeated eigenvalues.