Resolvent intertwining and spectral duality in Markov chains with geometric resetting

Abstract

We uncover the resolvent origin of the spectral duality governing reset-neutral distributions in Markov chains with geometric resetting. Starting from the abstract conditions of Paper~III, we show that the spectral duality $B_{\nu }(z)=\kappa (z)A_{\nu }(\sigma (z))$ is equivalent to a single symmetry of the resolvent $R(\gamma )=(I-(1-\gamma )P{)}^{-1}$: the intertwining relation $[{\Delta }^2\mathcal{R},R(\gamma )]=0$, where $\mathcal{R}$ is the reflection operator of an involution $\sigma$ and $\Delta =\mathrm{diag}(\sqrt{\kappa (z)})$; equivalently, $\tilde{\mathcal{T}}=K^{-1/2}{\Delta }^2\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^{\ast }=1/(1+\sqrt{K})$, with $K=\kappa (z)\kappa (\sigma (z))$, which depends only on the scalar $K$ --- not on the resetting rate $\gamma$, the reset distribution, or the particular chain. We characterize the class of $(\sigma ,\kappa )$-reversible chains, encompassing both the biased random walk and genuinely non-homogeneous dynamics sharing the same $C^{\ast }$; a Doob $h$-transform realizes the duality $K\mapsto 1/K$, hence $C^{\ast }\mapsto 1-C^{\ast }$, with fixed point $C^{\ast }=1/2$. The orientation field admits the explicit resolvent representation $\psi (\gamma )=R(\gamma )(b^{(0)}-C^{\ast }b)$: its gauge-normalized form ${\Delta }^{-1}\psi (\gamma )$ is antisymmetric under $\sigma$, it has an exact node at the fixed point of $\sigma$, and it governs the exact sign law $\mathrm{sgn}(C(\pi ,\gamma )-C^{\ast })=\mathrm{sgn}⟨\pi ,\psi (\gamma )⟩$. Numerical experiments confirm the theory to machine precision. These results establish the operator-theoretic foundation of the spectral duality of Paper~III and provide the bridge to the information-geometric framework of Paper~V.

17 pages, 3 figures, 5 tables