Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario
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Submission date: 23. Nov. 2017
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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)
We relate the expectation of the volume of 𝒮ℓ,n with some statistics of the smallest eigenvalue of a GOE(n) matrix, by providing explicit formulas for this quantity. These formulas imply that as ℓ,n →∞ on average 𝒮ℓ,n keeps a positive fraction of the volume of the sphere Sℓ (the exact constant is Φ(−1) ≈ 0.1587, where Φ is the cumulative distribution function of a standard gaussian variable).
For ℓ = 2 spectrahedra are generically smooth, but already when ℓ = 3 singular points on their boundaries appear with positive probability. We relate the average number 𝔼σn of singular points on the boundary of a three-dimensional spectrahedron 𝒮3,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 𝔼σ4 = 6 −. Moreover, we prove that the average number 𝔼ρn of singular points on the random symmetroid surface