Equi-energy sampling does not converge rapidly on the mean-field Potts model with three colors close to the critical temperature

Equi-energy sampling (EES, for short) is a method to speed up the convergence of the Metropolis chain, when the latter is slow. We show that there are still models like the mean-field Potts model, where EES does not converge rapidly in certain temperature regimes. Indeed we will show that EES is slowly mixing on the mean-field Potts model, in a regime below the critical temperature. Though we will concentrate on the Potts model with three colors, our arguments remain valid for any number of colors , if we adapt the temperature regime. For the situation of the mean-field Potts model this answers a question posed in Hua and Kou (2011 Stat. Sin. 21 1687–711). (paper)