论文标题
具有方差曲线（II）的非热随机矩阵：属性和示例
Non-Hermitian random matrices with a variance profile (II): properties and examples
论文作者
论文摘要
对于每个$ n $，让$ a_n =（σ_{ij}）$为$ n \ times n $确定矩阵，让$ x_n =（x__ {ij}）$为a $ n \ times n $随机矩阵，带有i.i.d.单位差异的中心条目。在伴侣文章Cook等人中，我们考虑了重新验证的入门产品的经验频谱分布$μ_n^y $ $ \ [y_n = \ frac 1 {\ sqrt {n}} a_n \ odot x_n = \ left（\ odot x_n = \ left（ σ_{ij} x_ {ij} \ right）\] \]，并提供了确定的概率测量序列$μ_n$，以使差异$μ^y_n -μ_n$的差异薄弱地收敛到零度量。 Cook等人的关键功能。是要允许某些条目$σ_{ij} $消失，前提是标准偏差配置文件$ a_n $满足某些定量不可保证属性。 在本文中，我们提供了有关序列$（μ_n）$的更多信息，该$由主方程式所描述。我们在重要特殊情况下考虑这些方程，例如可分离差异概况$σ^2_ {ij} = d_i \ widetilde d_j $和采样方差概况$σ^2_ {ij} =σ^2 \ 2 \ weft（\ frac in，\ frac jn \ right）$ y y y y y y y y y y y y y y y（y y y y y y y y y y y） $ [0,1]^2 $。提供$μ_n^y $收敛到真实限制的地方提供了关联示例。 我们以零研究$μ_n$的行为，并提供$μ_n$的密度有限，炸毁或消失的示例。结果，我们确定产生循环定律的曲线。 最后，基于Alt等人的最新结果，我们证明，除了零以外，$μ_n$承认在半径$ \ sqrt {ρ（v_n）} $的中心光盘上，$ v_n =（\ frac1nσ_{ij}^2）$ and $ρ（v_n）$是它的spectrals us的spectrals ins Spectrals ins Spection。
For each $n$, let $A_n=(σ_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. In the companion article Cook et al., we considered the empirical spectral distribution $μ_n^Y$ of the rescaled entry-wise product \[ Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}} σ_{ij}X_{ij}\right) \] and provided a deterministic sequence of probability measures $μ_n$ such that the difference $μ^Y_n - μ_n$ converges weakly in probability to the zero measure. A key feature in Cook et al. was to allow some of the entries $σ_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence $(μ_n)$, described by a family of Master Equations. We consider these equations in important special cases such as separable variance profiles $σ^2_{ij}=d_i \widetilde d_j$ and sampled variance profiles $σ^2_{ij} = σ^2\left(\frac in, \frac jn \right)$ where $(x,y)\mapsto σ^2(x,y)$ is a given function on $[0,1]^2$. Associate examples are provided where $μ_n^Y$ converges to a genuine limit. We study $μ_n$'s behavior at zero and provide examples where $μ_n$'s density is bounded, blows up, or vanishes while an atom appears. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al., we prove that except maybe in zero, $μ_n$ admits a positive density on the centered disc of radius $\sqrt{ρ(V_n)}$, where $V_n=(\frac 1n σ_{ij}^2)$ and $ρ(V_n)$ is its spectral radius.
