  We exhibit examples of high-dimensional unimodal posterior distributions
arising in non-linear regression models with Gaussian process priors for which
MCMC methods can take an exponential run-time to enter the regions where the
bulk of the posterior measure concentrates. Our results apply to worst-case
initialised (`cold start') algorithms that are local in the sense that their
step-sizes cannot be too large on average. The counter-examples hold for
general MCMC schemes based on gradient or random walk steps, and the theory is
illustrated for Metropolis-Hastings adjusted methods such as pCN and MALA.
