  We introduce the energy-stepping Monte Carlo (ESMC) method, a Markov chain
Monte Carlo (MCMC) algorithm based on the conventional dynamical interpretation
of the proposal stage but employing an energy-stepping integrator. The
energy-stepping integrator is quasi-explicit, symplectic, energy-conserving,
and symmetry-preserving. As a result of the exact energy conservation of
energy-stepping integrators, ESMC has a 100\%\ acceptance ratio of the proposal
states. Numerical tests provide empirical evidence that ESMC affords a number
of additional benefits: the Markov chains it generates have weak
autocorrelation, it has the ability to explore distant characteristic sets of
the sampled probability distribution and it yields smaller errors than chains
sampled with Hamiltonian Monte Carlo (HMC) and similar step sizes. Finally,
ESMC benefits from the exact symmetry conservation properties of the
energy-stepping integrator when sampling from potentials with built-in
symmetries, whether explicitly known or not.
