Local and Global Contraction Principles for MCMC Mixing

We develop a contraction-based framework for proving mixing-time bounds for Markov chain Monte Carlo algorithms. The framework is built around global and local contraction coefficients of Markov kernels under the mathsf Eγ-divergence with γ≥1. For projected Langevin Monte Carlo on a compact convex domain, we show that Gaussian smoothing yields an explicit global contraction coefficient for the mathsf Eγ-divergence. This gives a direct proof of exponential convergence to the discretized stationary distribution for general smooth, possibly non-convex potentials. The rate is explicit, accommodates arbitrary random-batch sampling schemes, and yields convergence guarantees for several divergences, including KL, chi2, and R'enyi divergences. For independent Metropolis--Hastings with target π, proposal q, and unbounded importance weight w=dπ/dq, global contraction coefficients are typically trivial. We therefore introduce a local contraction coefficient on the core CR=w≤ R and prove that it controls the rejection profile on the core. This yields warm-start convergence bounds governed by the local contraction coefficient and the tail profile HR=π(w>R), recovering sharp existing moment-based convergence rates when mathbb Eq[wp]<∞ for some p>1, while remaining effective in heavy-tailed regimes where no finite moment of order p>1 exists.