Rigorous lower bound on the dynamic critical exponents of the Swendsen-Wang algorithm

We prove the rigorous lower bound $z_{\mathrm{sw}\ge \mathrm{\alpha }/\nu }$ for the dynamic critical exponent of the Swendsen-Wang algorithm. For two-dimensional q-state Potts models with q=2,3,4, this implies $z_{\mathrm{sw}\ge 0}$,(2/5,1. We present numerical data indicating that $z_{\mathrm{sw}=0.55\mathrm{}\pm 0.03}$, 0.89±0.05 for q=3,4 (95% confidence limits, statistical errors only). The discrepancy for q=4 appears to be caused by multiplicative logarithmic corrections.