On the Spectral Region of n-Cycle Stochastic Matrices

Abstract

For every $n\geq3$, we determine the union of the spectra of row-stochastic matrices supported on a directed $n$-cycle with self-loops and strictly positive cycle edges. These matrices describe progression through cyclic stages with geometric waiting times, and also arise by uniformization of unidirectional continuous-time reaction cycles. For the latter, the classification determines the minimum transition-rate bound required to realize a prescribed nonreal eigenvalue and provides rates attaining that bound. The angles $m=\mathrm{Arg}λ$ and $M=\mathrm{Arg}(λ-1)$ reduce the nonreal eigenvalue problem to determining the range of a strictly convex logarithmic sum over angles with prescribed sum. Jensen's inequality gives its minimum, simplex vertices give its finite maximum, and connectedness makes these bounds sufficient for realization. The resulting branch graphs are strictly decreasing and their horizontal sections are ordered. This ordering gives the complete boundary, which alternates between uniform-cycle segments and one-loop algebraic arcs. We state a direct membership criterion and construct a realizing matrix for every spectral point; every nonreal point admits a realization with at most two distinct self-loop weights. The real spectral set is $[-1,1]$ for even $n$ and $(0,1]$ for odd $n$. Allowing deleted cycle edges adds only $0$ in odd dimension and adds nothing in even dimension. We also apply the classification to a branching network with matched holding probabilities on parallel paths. The proofs are independent of Karpelevich's theorem.