The Single Ring Theorem and a Question of Shub

2025/09/12 by Paik, Joshua
#37 #60 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2509.10358

Abstract

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 c_d,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, c_d,k → 1 as d → ∞, and k = 1.

Citations

• Local single ring theorem on optimal scale
• Support convergence in the single ring theorem
• The single ring theorem
• On some mean matrix inequalities of dynamical interest

Related

• The single ring theorem
• Local single ring theorem on optimal scale
• Fluctuations for analytic test functions in the Single Ring Theorem
• Deformed single ring theorems
• Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle