Many popular Markov chain Monte Carlo samplers for statistical mechanics models drastically slow down in the vicinity of a phase transition. This slowdown is deeply tied to the geometry of the model. I will discuss some aspects of this phenomenon and show in particular that, on certain geometries, it is possible to identify the cause(s) of the slowdown. This can then be used to build a physically reasonable dynamics, which samples from the model from which slow objects have been removed in a sense I will make precise, and for which fast sampling guarantees can be obtained beyond the phase transition. This is a joint work with Roland Bauerschmidt and Thierry Bodineau based on the following paper: https://link.springer.com/article/10.1007/s00440-024-01326-9 .