The Single Ring Theorem and a Question of Shub

Abstract: 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 $c_{d,k} \to 0 as d \to \infty$. 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 $d \to \infty$, and $k = 1$.