While a random walk Metropolis-Hastings algorithm cannot be uniformly ergodic in a general setting (Mengersen and Tweedie, AoS, 1996), because it needs more energy to leave far away starting points, it can be geometrically ergodic depending on the target (and the proposal). In a recent Annals of Statistics paper, Leif Johnson and Charlie Geyer designed a trick to turn a random walk Metropolis-Hastings algorithm into a geometrically ergodic random walk Metropolis-Hastings algorithm by virtue of an isotropic transform (under the provision that the original target density has a moment generating function). This theoretical result is complemented by an R package called mcmc. (I have not tested it so far, having read the paper in the métro.) The examples included in the paper are however fairly academic and I wonder how the method performs in practice, on truly complex models, in particular because the change of variables relies on (a) an origin and (b) changing the curvature of space uniformly in all dimensions. Nonetheless, the idea is attractive and reminds me of a project of ours with Randal Douc,  started thanks to the ‘Og and still under completion.