Preprint 76/2017

Random Spectrahedra

Paul Breiding, Khazhgali Kozhasov, and Antonio Lerario

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Submission date: 23. Nov. 2017
Pages: 24
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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)

𝒮 ℓ,n = {(x0,...,xℓ) ∈ Sℓ | x0I + x1R1 + ⋅⋅⋅+ xℓRℓ ≻ 0}
where R1,,R are independent GOE(n)-distributed matrices rescaled by (2nℓ)12.

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 S (the exact constant is Φ(1) 0.1587, where Φ is the cumulative distribution function of a standard gaussian variable).

For = 2 spectrahedra are generically smooth, but already when = 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 -4- √3-. Moreover, we prove that the average number 𝔼ρn of singular points on the random symmetroid surface

                       3 Σ3,n = {(x0,x1,x2,x3) ∈ S | det(x0I + x1R1 +x2R2 + x3R3) = 0},
equals n(n 1). This quantity is related to the volume of the set of symmetric matrices with repeated eigenvalues.

02.12.2017, 01:43