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 $A_n(α)$ for the matrix indexed by $\mathbb Z/n\mathbb Z$ with $$ (A_n(α))_{j,j}=α_j,\qquad (A_n(α))_{j,j+1}=1-α_j,\qquad 0\le α_j<1, $$ and all other entries zero, and set $C_n=\{A_n(α):α\in[0,1)^n\}$. Writing $Σ_n$ for the corresponding spectral union, the trivial cases are $Σ_1=\{1\}$ and $Σ_2=[-1,1]$. For $n\ge 3$, we give an explicit description of $Σ_n$ in angular coordinates $m=\mathrm{Arg}(λ)$ and $M=\mathrm{Arg}(λ-1)$. Under the map $$ Λ(m,M)=\frac{\sin M}{\sin(M-m)}e^{im}, $$ the upper half of $Σ_n$ 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 $M=ϕ_k$, 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&#39;s theorem and reduces the two-monomial characteristic equation to sharp argument bounds on a simplex, obtained by Jensen, majorization, and finite visibility arguments.