  We analyze the so called Swapping Algorithm, a parallel version of the
well-known Metropolis-Hastings algorithm, on the mean-field version of the
Blume-Emery-Griffiths model in statistical mechanics. This model has two
parameters and depending on their choice, the model exhibits either a first, or
a second order phase transition. In agreement with a conjecture by Bhatnagar
and Randall we find that the Swapping Algorithm mixes rapidly in presence of a
second order phase transition, while becoming slow when the phase transition is
first order.
