Metropolis algorithm and equienergy sampling for two mean field spin systems

In this paper we study the Metropolis algorithm in connection with two mean–field spin systems, the so called mean–field Ising model and the Blume–Emery–Griffiths model. In both this examples the naive choice of proposal chain gives rise, for some parameters, to a slowly mixing Metropolis chain, that is a chain whose spectral gap decreases exponentially fast (in the dimension $N$ of the problem). Here we show how a slight variant in the proposal chain can avoid this problem, keeping the mean computational cost similar to the cost of the usual Metropolis. More precisely we prove that, with a suitable variant in the proposal, the Metropolis chain has a spectral gap which decreases polynomially in 1/N. Using some symmetry structure of the energy, the method rests on allowing appropriate jumps within the energy level of the starting state.