Cost of the generalised hybrid Monte Carlo algorithm for free field theory

We study analytically the computational cost of the generalised hybrid Mont e Carlo (GHMC) algorithm for free field theory. We calculate the Metropolis acceptance probability for leapfrog and higher-order discretisations of th e molecular dynamics (MD) equations of motion. We show how to calculate aut ocorrelation functions of arbitrary polynomial operators, and use these to optimise the GHMC momentum mixing angle. the trajectory length, and the int egration stepsize for the special cases of linear and quadratic operators. We show that long trajectories are optimal for GHMC, and that standard HMC is more efficient than algorithms based on second order Langevin Monte Carl o (L2MC), sometimes known as Kramers equation. We show that contrary to nai ve expectations HMC and L2MC have the same volume dependence, but their dyn amical critical exponents are z = 1 and z = 3/2, respectively. (C) 2001 Els evier Science B.V. All rights reserved.