Kinematics of Multigrid Monte Carlo

TL;DR

This paper analyzes the efficiency of multigrid Monte Carlo algorithms by deriving an acceptance rate approximation, comparing interpolation schemes, and applying the method to gauge theories, revealing conditions to avoid critical slowing down.

Contribution

It introduces an approximation formula for acceptance rates, compares interpolation schemes in free field theory, and extends multigrid updates to nonabelian gauge theories.

Findings

Acceptance rate approximation formula matches simulations.

Absence of critical slowing down linked to the Hamiltonian expansion.

Multigrid updates are effective for SU(2) gauge theory.

Abstract

We study the kinematics of multigrid Monte Carlo algorithms by means of acceptance rates for nonlocal Metropolis update proposals. An approximation formula for acceptance rates is derived. We present a comparison of different coarse-to-fine interpolation schemes in free field theory, where the formula is exact. The predictions of the approximation formula for several interacting models are well confirmed by Monte Carlo simulations. The following rule is found: For a critical model with fundamental Hamiltonian H(phi), absence of critical slowing down can only be expected if the expansion of <H(phi + psi)> in terms of the shift psi contains no relevant (mass) term. We also introduce a multigrid update procedure for nonabelian lattice gauge theory and study the acceptance rates for gauge group SU(2) in four dimensions.