发表机构
Vrije Universiteit Brussel (VUB); imec-SMIT; Harvard University（布鲁塞尔自由大学; imec-SMIT; 哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文精确刻画双层更新随机矩阵族的谱并集，通过等谱平衡映射和转移几何将其约化为单纯形上的三角面实现，并证明边界由Farey算术选择的边载体唯一生成，且谱区域径向填充并具有特定扇形缺口。
AI 中文摘要
固定 $q\ge2$，考虑由两条确定性 $q$ 步路径构成的 $2q\times2q$ 行随机矩阵，其随机性仅限于终行。我们精确确定了该族的谱并集。一个等谱平衡映射将全参数空间约化为 $q$-单纯形，而转移几何为每个谱点提供了三角面实现。可见边界更为微妙：Farey 算术在单纯形一维骨架上选择一条游走，其底边和侧边提供边界载体。在侧边情形中，平方边方程有两个代数平方根，但仅扇形选择的分支是支撑的。我们证明所选载体在每条射线上是唯一的且最外侧，且谱区域在其下方径向填充。该区域的实截面为 $[-1,1]$，与单位圆恰好交于阶数至多 $2q$ 的单位根，并省略扇形 $0<|\operatorname{Arg}\lambda|<\pi/q$。对于奇数 $q$，终边还贡献一个负实边界区间。因此，三角面生成谱并集，而单纯形边生成其边界。
英文摘要
We determine the complete eigenvalue region of the following Markov chains on $2q$ states, for $q\ge2$. The states are $A_0,\ldots,A_{q-1}$ and $B_0,\ldots,B_{q-1}$. Each transition $A_j\to A_{j+1}$ and $B_j\to B_{j+1}$, for $0\le j<q-1$, has probability one. From $A_{q-1}$, the chain moves to $A_0$ with probability $a$ and to $B_0$ with probability $1-a$. From $B_{q-1}$, it moves to $B_0$ with probability $b$ and to $A_j$ with probability $(1-b)p_j$. Here $a,b\in[0,1]$ and $p_j\ge0$ with $\sum_{j=0}^{q-1}p_j=1$; these parameters vary over all permitted values. Every segment from zero to an attainable eigenvalue lies in the region. Its unit-circle points are exactly the roots of unity of order at most $2q$, as for unrestricted stochastic matrices of the same order, yet the full region is strictly smaller. It excludes $0<|\operatorname{Arg}λ|<π/q$. For odd $q$, the final nonreal boundary arc ends at a negative real point inside the unit disk; the remaining interval to $-1$ is also boundary. Every transition matrix has the same characteristic polynomial as one with $a=b$. Among $q+1$ fixed matrices in this balanced family, convex combinations of at most three realize every attainable eigenvalue, and two suffice on the boundary. Both bounds are sharp within this set. Consecutive reduced fractions in $[1/(2q),1/2]$ with denominators at most $2q$ select the upper nonreal boundary constructions. Each boundary radius is the unique solution of an equation whose left-hand side increases strictly with the radius.
Comments48 pages, 6 figures, 1 table
