论文标题
用于二次多项式的光谱和伪谱。
Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices
论文作者
论文摘要
对于固定的二次多项式$ \ mathfrak {p} $ in $ n $非通知变量，以及$ n $ nepplion $ n \ times n $ n $复杂的ginibre $ x_1^n，\ dots，x_n^n $ x_n^n）$ to $ \ mathfrak {p} $在$ n $自由独立的循环元素上评估的$ \ mathfrak {p} $ $ c_1，\ dots，c_n $在非交换概率空间中。证明的主要步骤是在$ p^n $的伪谱上获得定量控制。通过众所周知的线性化欺骗，这取决于某些基质值随机步行的抗浓缩特性，我们发现这可能会出于与算术性障碍物不同的结构原因而失败，这些结构原因与在Littlewood-Offord - Offord问题中所阐明的算术障碍物不同。
For a fixed quadratic polynomial $\mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =\mathfrak{p}(X_1^N,\dots, X_n^N)$ to the Brown measure of $\mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,\dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.
