Empirical relations between static and dynamic exponents for Ising model cluster algorithms

We have measured the autocorrelations for the Swendsen-Wang and the Wolff cluster update algorithms for the Ising model in two, three, and four dimensions. The data for the Wolff algorithm suggest that the autocorrelations are linearly related to the specific heat, in which case the dynamic critical exponent is ${\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$ ${\mathit{E}}^{\mathrm{W}}$=α/ν. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that ${\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$ ${\mathit{E}}^{\mathrm{SW}}$=β/ν for the Ising model.