Mean-field behavior of cluster dynamics

The dynamic behavior of cluster algorithms is analyzed in the classical mean-field limit. Rigorous analytical results below [Formula Presented] establish that the dynamic exponent has the value [Formula Presented]=1 for the Swendsen-Wang algorithm and [Formula Presented]=0 for the Wolff algorithm. An efficient Monte Carlo implementation is introduced, adapted for using these algorithms for fully connected graphs. Extensive simulations both above and below [Formula Presented] demonstrate scaling and evaluate the finite-size scaling function by means of a rather impressive collapse of the data.