随机矩阵比值的硬边界行列式场与二阶矩普适性
Hard-Edge Determinant Fields and Second-Moment Universality for Random Matrix Ratios
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中文总结 AI 辅助
该研究针对独立复Girko矩阵比值，证明其归一化行列式的紧一致收敛性，得到普适极限，回答了二阶矩相关问题，并扩展了结论至原子律三角阵列。
中文摘要 AI 辅助
我们研究独立复Girko矩阵比值在原点处的微观谱。在有界密度、有限矩假设及二阶矩与圆律匹配的条件下，我们证明了归一化行列式的紧一致收敛性，其极限为从复硬边界与独立Ginibre阵列构造的随机整函数。该极限行列式场的零点集服从无限复Ginibre过程的分布。因此，前有限个最小和最大特征值模、内外谱半径以及固定有界集内的特征值计数均具有仅依赖二阶矩的普适极限，这回答了Chafaï、García-Zelada和Xu针对谱半径提出的二阶矩问题。我们还得到了针对有限个扰动方向的多变量行列式场，并证明了一致硬边界比较，该比较可将结论扩展至原子律的三角阵列。
英文摘要
We study the microscopic spectrum at the origin of ratios of independent complex Girko matrices. Under bounded-density and finite-moment assumptions, together with circular second-moment matching, we prove compact-uniform convergence of the normalized determinants to a random entire function constructed from the complex hard edge and an independent Ginibre array. The zero divisor of this limiting determinant field has the law of the infinite complex Ginibre process. Consequently, the first finitely many smallest and largest eigenvalue moduli, the inner and outer spectral radii, and eigenvalue counts in fixed bounded sets have universal limits depending only on the second moments. This answers the second-moment question posed by Chafaï, García-Zelada, and Xu for the spectral radii. We also obtain a multivariate determinant field for finitely many perturbation directions and prove the uniform hard-edge comparison needed to extend the conclusions to triangular arrays of atom laws.
