The Wang-Landau algorithm reaches the Flat Histogram criterion in finite time

The Wang-Landau algorithm aims at sampling from a probability distribution, while penalizing some regions of the state space and favouring others. It is widely used, but its convergence properties are still unknown. We show that for some variations of the algorithm, the Wang-Landau algorithm reaches the so-called Flat Histogram criterion in finite time, and that this criterion can be never reached for other variations. The arguments are shown on an simple context -- compact spaces, density functions bounded from both sides-- for the sake of clarity, and could be extended to more general contexts.