Efficiency of Rejection-free dynamic Monte Carlo methods for homogeneous spin models, hard disk systems, and hard sphere systems

We construct asymptotic arguments for the relative efficiency of rejection-free Monte Carlo (MC) methods compared to the standard MC method. We find that the efficiency is proportional to $exp{({const} beta)}$ in the Ising, $sqrt{beta}$ in the classical XY, and $beta$ in the classical Heisenberg spin systems with inverse temperature $beta$, regardless of the dimension. The efficiency in hard particle systems is also obtained, and found to be proportional to $(rhoc -rho)^{-d}$ with the closest packing density $rhoc$, density $rho$, and dimension $d$ of the systems. We construct and implement a rejection-free Monte Carlo method for the hard-disk system. The RFMC has a greater computational efficiency at high densities, and the density dependence of the efficiency is as predicted by our arguments.