  Among random sampling methods, Markov Chain Monte Carlo algorithms are
foremost. Using a combination of analytical and numerical approaches, we study
their convergence properties towards the steady state, within a random walk
Metropolis scheme. Analysing the relaxation properties of some model algorithms
sufficiently simple to enable analytic progress, we show that the deviations
from the target steady-state distribution can feature a localization transition
as a function of the characteristic length of the attempted jumps defining the
random walk. While the iteration of the Monte Carlo algorithm converges to
equilibrium for all choices of jump parameters, the localization transition
changes drastically the asymptotic shape of the difference between the
probability distribution reached after a finite number of steps of the
algorithm and the target equilibrium distribution. We argue that the relaxation
before and after the localisation transition is respectively limited by
diffusion and rejection rates.
