Eigenvalue bounds on restrictions of reversible nearly uncoupled Markov chains
The authors analyze decompositions of reversible nearly uncoupled Markov chains into rapid mixing subchains, presenting upper bounds on the second eigenvalue for restriction and stochastic complementation chains of reversible Markov chains as well as a relation between them. The authors also illustrate the obtained bounds analytically for bunkbed graphs, and apply the to restricted Markov chains that arise when analyzing comformation dynamics of a small biomolecule. The efficiency of applying an uncoupling-coupling technique to a nearly uncoupled Markov chain essentially depends on the second eigenvalues of the resulting subchain which is bounded far away from \(1\). The main theorems of this article provide bounds on this spectral gap. The theorem may help better understand the structure and similarities of uncoupled chains to the original Markov chain and thus shed some light on the usefulness and applicability of uncoupling-coupling techniques.
scientific article; zbMATH DE number 2165377