Norms of structured random matrices.

For , let be a random matrix, a real deterministic matrix, and the corresponding structured random matrix. We study the expected operator norm of considered as a random operator between and for . We prove optimal bounds up to logarithmic terms when the underlying random matrix has i.

The Minimal Spherical Dispersion.

We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy in (Anz Österreich Akad Wiss Math Nat Kl 132:3-10, 1995). In particular, we see that the inverse of the minimal spherical dispersion is, for fixed , linear in the dimension of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere.