Can be very difficult to prove that the method of choosing new
configurations is ergodic (especially at 
).
Can get trapped in subsets of configuration space, such as periodic loops.
May also have updates that are quasi-ergodic --- there may be a high energy barrier between subspaces, so it is theoretically possible to reach all possible states, but highly improbable in a finite time simulation.
Check: Compare results from multiple runs with different initial configurations.