临界不变多边形与随机谱的Farey–Ito边界
A proof of the Karpelevič theorem via invariant polygons
- Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）
- imec-SMIT
- Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过发展径向临界映射的接触理论，利用Farey邻域和Ito积结构，完整刻画了实数行随机矩阵谱并集的Farey–Ito边界，并证明了嵌套性。
AI中文摘要：
设$T$为保向平面（带定向）上的实线性压缩映射，其特征值非实数，其多边形复杂度定义为满足$TP\subseteq P$的非退化凸多边形$P$的最少顶点数。我们首先为径向临界映射发展了一套内蕴的接触理论。遗传饱和性、单侧接触选择、精确接触突变、有限循环约化以及幺模格帆导出了接触-回归规范形和射影无跳跃原理。在适应的复坐标下，所得单值化产生一个异质Ito积以及由Farey邻域承载的精确提升相位恒等式。随后，我们将此结构应用于确定所有实数行随机$n\times n$矩阵谱的并集$\Theta_n$（对每个$n$）。一个严格凸的对数正弦势使单值化因子相等，并给出尖锐的径向方程。稀疏随机矩阵达到每个等式点，而标量Farey细化证明了从$n-1$阶到$n$阶的嵌套性。结合径向填充和单位圆分析，这从临界不变多边形建立了Karpelevič区域的完整Farey–Ito描述。
英文摘要：
The Karpelevič theorem describes the complex numbers that occur as eigenvalues of stochastic matrices of a fixed order. We give a self-contained proof based on invariant polygons. Their geometry yields a product equation for an eigenvalue of maximal modulus at a fixed argument. Convexity determines this modulus, explicit stochastic matrices attain it, and a comparison between Farey intervals completes the boundary description. As the matrix order increases, the resulting equation gives a uniform relative asymptotic for the gap $1-|z|$ between a boundary point $z$ and the unit circle, even arbitrarily close to arc endpoints. We also determine the optimal contraction factor for a planar rotation and dilation when vector size is measured by scaling a polygon with at most $N$ vertices.
