Overview
This research investigates the distribution of eigenvalues for Haar-random matrices over $\mathbb{Z}_p$ within the framework of algebraic extensions of $\mathbb{Q}_p$. It also examines analogous distributions for roots of random Haar polynomials over $\mathbb{Z}_p$. The study draws comparisons to established real-eigenvalue and real-root counting results but identifies distinct behavior in the $p$-adic context.
Research Context
The study situates itself in the context of eigenvalue distribution for random matrices, specifically drawing parallels to the real-eigenvalue counting results by Edelman-Kostlan-Shub for the real Ginibre ensemble. For random polynomials, it refers to the real-root counting results of Edelman-Kostlan. The work by Caruso (arXiv:2110.03942) provides correlation function formulas for roots of random Haar polynomials, which are utilized. The author's prior joint work with Van Peski (arXiv:2601.06283) also contributes correlation function formulas relevant to this study.
Approach
The research approach involves studying Haar-random matrices over $\mathbb{Z}_p$ and random Haar polynomials over $\mathbb{Z}_p$. The distribution of eigenvalues and roots is analyzed over algebraic extensions of $\mathbb{Q}_p$. The proof methodologies incorporate correlation function formulas derived from the author's previous joint work with Van Peski (arXiv:2601.06283). Additionally, correlation function formulas by Caruso (arXiv:2110.03942) are applied for the polynomial analysis. A key aspect of the methodology involves uniform estimates across varying finite extensions.
Findings
- Eigenvalues of Haar-random matrices over $\mathbb{Z}_p$ exhibit an asymptotic even distribution among possible extension degrees when considered within algebraic extensions of $\mathbb{Q}_p$.
- This behavior contrasts with the real Ginibre ensemble, where real eigenvalues constitute a vanishing proportion.
- The maximal unramified extension $\mathbb{Q}_p^{\mathrm{un}}$ captures all but a bounded expected number of eigenvalues.
- The expected number of eigenvalues falling outside $\mathbb{Q}_p^{\mathrm{un}}$ has a finite positive limit.
- An explicit upper bound has been established for this finite positive limit of eigenvalues outside $\mathbb{Q}_p^{\mathrm{un}}$.
- Analogous results were observed for roots of random Haar polynomials over $\mathbb{Z}_p$.
- These polynomial results represent $p$-adic analogues of the real-root counting results by Edelman-Kostlan, also demonstrating different behavior compared to the real setting.