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

TL;DR

This paper investigates finite-time spectral rigidity in reversible Markov chains, revealing new spectral structures, entropy dynamics, and convergence criteria that extend beyond classical asymptotic analysis.

Contribution

It introduces explicit bounds on spectral rigidity time, develops a spectral entropy theory, and provides data-driven stopping criteria for power iteration in reversible Markov chains.

Findings

Explicit bounds on rigidity time based on spectral separation ratio.

Spectral entropy attains maximum at half-rigidity threshold with sign change in covariance.

Data-driven adaptive stopping criterion with provable guarantees.

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 $L^2(\pi)$, organizing the dynamics around the relaxation operator $\mathcal{G}=I-P$ reveals finite-time structures invisible to traditional asymptotic estimates. For chains with $\lambda_2>\lambda_3$, we establish explicit two-sided bounds on the rigidity time $T_{\mathrm{rigid}}(\delta)$, the first moment the slowest mode captures a fraction $1-\delta$ 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 $\lambda_3/\lambda_2$, not the classical gap $1-\lambda_2$ alone. We develop a spectral entropy theory governed by the exact balance law $\Delta S=\mathrm{Cov}/\rho-D_{\mathrm{KL}}$ 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 $\mathrm{Var}_{p_k}[\lambda^2]=\rho_k(\rho_{k+1}-\rho_k)$, 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.