What type of study is this?

This is a Quantitative Study study.

September 30, 2025Open Access

The Single Ring Theorem and a Question of Shub

Key Points

The research demonstrates that the conjecture holds asymptotically for the spectral radius as dimensions increase.
A key finding is that the constant converges to 1 as the dimension approaches infinity for the case of k = 1.
The study builds on the previously established Single Ring Theorem to provide a more precise relationship.
It highlights the difference in behavior of eigenvalues and distortion metrics along Grassmanians.

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 c₃, ₊ 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, c₃, ₊ 1 as d, and k = 1.

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Cite This Study

Joshua Paik (Fri,) studied this question.

synapsesocial.com/papers/68dc1e358a7d58c25ebb1903 https://doi.org/https://doi.org/10.48550/arxiv.2509.10358

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