The Single Ring Theorem and a Question of Shub

Given an orthogonally invariant probability measure on GL(d,R), Mike Shub asked whether the average product of the k top eigenvalues in the ensemble can be lower bounded by the average distortion along k dimensional Grassmanians. Recently, Armentano, Chinta, Sahi, and Shub provided partial progress, however they attach a constant cd,k → 0 as d → ∞. In this paper, by invoking the Single Ring Theorem and sequels, we show the conjecture asymptotically for the spectral radius, in particular, cd,k → 1 as d → ∞, and $k = 1$.