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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.10358 |
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| _version_ | 1866914040853299200 |
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| author | Paik, Joshua |
| author_facet | Paik, Joshua |
| contents | 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 $k = 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Single Ring Theorem and a Question of Shub Paik, Joshua Probability Dynamical Systems 37, 60 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 $k = 1$. |
| title | The Single Ring Theorem and a Question of Shub |
| topic | Probability Dynamical Systems 37, 60 |
| url | https://arxiv.org/abs/2509.10358 |