Cost of Generalised HMC algorithms 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 autocorrel ation functions of operators quadratic in the fields, and optimise the GHMC momentum mixing angle, the trajectory length, and the integration stepsize . We show that long trajectories are optimal for GHMC, and that standard HM C is much more efficient than algorithms based on the Second Order Langevin (L2MC) or Kramers Equation. We show that contrary to naive expectations HM C and L2MC have the same volume dependence, but their dynamical critical ex ponents are z = 1 and z = 3/2 respectively.