The Single Ring Theorem and a Question of Shub

Given an orthogonally invariant probability measure on $GL(d,\mathbb{R})$, 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 $c_{d,k}\to 0asd\to \mathrm{\infty }$. In this paper, by invoking the Single Ring Theorem and sequels, we show the conjecture asymptotically for the spectral radius, in particular, $c_{d,k}\to 1$ as $d\to \mathrm{\infty }$, and $k=1$.