The Swendsen-Wang process does not always mix rapidly

The Swendsen-Wang process provides one possible dynamics for the q-state Potts model. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The legitimacy of such simulations depends on the rate of convergence of the process to equilibrium, as measured by the "mixing time." Empirical observations suggest that the mixing time of the Swendsen-Wang process is short in many instances of practical interest, although proofs of this desirable behavior are known only for some very special cases. Nevertheless, we show that there are occasions when the mixing time of the Swendsen-Wang process is exponential in the size of the system. This undesirable behavior is related to the phenomenon of first-order phase transitions in Potts systems with q>2 states.