  This paper surveys in detail the relations between numerical integration and
the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational
cost of HMC mainly lies in the numerical integrations, these should be
performed as efficiently as possible. However, HMC requires methods that have
the geometric properties of being volume-preserving and reversible, and this
limits the number of integrators that may be used. On the other hand, these
geometric properties have important quantitative implications on the
integration error, which in turn have an impact on the acceptance rate of the
proposal. While at present the velocity Verlet algorithm is the method of
choice for good reasons, we argue that Verlet can be improved upon. We also
discuss in detail the behavior of HMC as the dimensionality of the target
distribution increases.
