  We study the problem of sampling high and infinite dimensional target
measures arising in applications such as conditioned diffusions and inverse
problems. We focus on those that arise from approximating measures on Hilbert
spaces defined via a density with respect to a Gaussian reference measure. We
consider the Metropolis-Hastings algorithm that adds an accept-reject mechanism
to a Markov chain proposal in order to make the chain reversible with respect
to the target measure. We focus on cases where the proposal is either a
Gaussian random walk (RWM) with covariance equal to that of the reference
measure or an Ornstein-Uhlenbeck proposal (pCN) for which the reference measure
is invariant. Previous results in terms of scaling and diffusion limits
suggested that the pCN has a convergence rate that is independent of the
dimension while the RWM method has undesirable dimension-dependent behaviour.
We confirm this claim by exhibiting a dimension-independent Wasserstein
spectral gap for pCN algorithm for a large class of target measures. In our
setting this Wasserstein spectral gap implies an $L^2$-spectral gap. We use
both spectral gaps to show that the ergodic average satisfies a strong law of
large numbers, the central limit theorem and nonasymptotic bounds on the mean
square error, all dimension independent. In contrast we show that the spectral
gap of the RWM algorithm applied to the reference measures degenerates as the
dimension tends to infinity.
