Quantitative Spectral Rigidity and Finite-Time Spectral Thermodynamics in Reversible Markov Chains

Abstract

We study finite-time spectral rigidity in reversible Markov chains via exact spectral relaxation dynamics. While the underlying identities follow classically from self-adjointness on $L^2(π)$, organizing the dynamics around the relaxation operator $\mathcal{G}=I-P$ reveals finite-time structures invisible to traditional asymptotic estimates. For chains with $λ_2>λ_3$, we establish explicit two-sided bounds on the rigidity time $T_{\mathrm{rigid}}(δ)$, the first moment the slowest mode captures a fraction $1-δ$ of the total spectral energy. The bounds differ by at most one step and show that rigidity emergence is controlled by the spectral separation ratio $λ_3/λ_2$, not the classical gap $1-λ_2$ alone. We develop a spectral entropy theory governed by the exact balance law $ΔS=\mathrm{Cov}/ρ-D_{\mathrm{KL}}$ and a canonical covariance representation of entropy transfer. In the two-mode case, the covariance changes sign precisely at the half-rigidity threshold $T_{\mathrm{rigid}}(1/2)$, where spectral entropy attains its maximum $\log 2$. For general chains, we obtain a sharp sufficient rigidity criterion for monotone entropy decay. Applied to power iteration, the framework yields an exact error identity, the observable spectral variance formula $\mathrm{Var}_{p_k}[λ^2]=ρ_k(ρ_{k+1}-ρ_k)$, and a fully data-driven adaptive stopping criterion with provable guarantees. These results demonstrate that reversible Markov chains possess a precise finite-time rigidity structure governing spectral purification, entropy dynamics, and observable convergence beyond classical asymptotic theory.