Solidarity of Gibbs Samplers: the spectral gap

Gibbs samplers are preeminent Markov chain Monte Carlo algorithms used in computational physics and statistical computing. Yet, their most fundamental properties, such as relations between convergence characteristics of their various versions, are not well understood. In this paper we prove the solidarity of their spectral gaps: if any of the random scan or $d!$ deterministic scans has a~spectral gap then all of them have. Our methods rely on geometric interpretation of the Gibbs samplers as alternating projection algorithms and analysis of the rate of convergence in the von Neumann--Halperin method of cyclic alternating projections. In addition, we provide a quantitative result: if the spectral gap of the random scan Gibbs sampler scales polynomially with dimension, so does the spectral gap of any of the deterministic scans.