Efficiency of rejection free Monte Carlo methods

We construct asymptotic arguments for the waiting time, the number of trials to achieve one update in Monte Carlo (MC) methods. We find that the waiting time is proportional to $\mbox{e}^\beta$ 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 behavior in hard particle systems is also obtained, and found to be proportional to $(\rho c -\rho)^{-d}$ with the closest packing density $\rho c$, density $\rho$, and dimension $d$ of the systems. The waiting time determines the efficiency of rejection-free Monte Carlo (RFMC) methods, as well as the inefficiency of the standard MC. Our arguments are general, and applicable to any RFMC implementation of any model studied using kinetic Monte Carlo.