On the Spectral Region of n-Cycle Stochastic Matrices

Abstract

For every $n$, we determine the complete eigenvalue region of the $n$-cycle stochastic family. For $n\ge 2$, write $A_n(\alpha)$ for the matrix indexed by $\mathbb Z/n\mathbb Z$ with $$ (A_n(\alpha))_{j,j}=\alpha_j,\qquad (A_n(\alpha))_{j,j+1}=1-\alpha_j,\qquad 0\le \alpha_j<1, $$ and all other entries zero, and set $C_n=\{A_n(\alpha):\alpha\in[0,1)^n\}$. Writing $\Sigma_n$ for the corresponding spectral union, the trivial cases are $\Sigma_1=\{1\}$ and $\Sigma_2=[-1,1]$. For $n\ge 3$, we give an explicit description of $\Sigma_n$ in angular coordinates $m=\mathrm{Arg}(\lambda)$ and $M=\mathrm{Arg}(\lambda-1)$. Under the map $$ \Lambda(m,M)=\frac{\sin M}{\sin(M-m)}e^{im}, $$ the upper half of $\Sigma_n$ is the image of a finite union of $K=\lfloor(n-1)/2\rfloor$ vertical angular sectors. Its exposed boundary is an alternating chain of Jensen chords, arising from the Jensen-equality lines $M=\phi_k$, and algebraic one-loop arcs joining the relevant roots of unity to $0$; the lower boundary is obtained by complex conjugation. The real spectral part is $[-1,1]$ for even $n$ and $(0,1]$ for odd $n$. The proof is independent of Karpelevich's theorem and reduces the two-monomial characteristic equation to sharp argument bounds on a simplex, obtained by Jensen, majorization, and finite visibility arguments.