Acceptance rates in multigrid Monte Carlo simulations

An approximation formula is derived for acceptance rates of nonlocal Metropolis updates in simulations of lattice field theories. The predictions of the formula agree quite well with Monte Carlo simulations of two-dimensional sine-Gordon, $\mathrm{XY}$, and ${\phi }^4$ models. The results are consistent with the following rule: For a critical model with a fundamental Hamiltonian $\mathcal{H}\left(\phi \right)$ sufficiently high acceptance rates for a complete elimination of critical slowing down can only be expected if the expansion of $〈\mathcal{H}(\phi +\psi )〉$ in terms of the shift $\psi$ contains no relevant term (mass term).