  We establish general conditions under which Markov chains produced by the
Hamiltonian Monte Carlo method will and will not be geometrically ergodic. We
consider implementations with both position-independent and position-dependent
integration times. In the former case we find that the conditions for geometric
ergodicity are essentially a gradient of the log-density which asymptotically
points towards the centre of the space and grows no faster than linearly. In an
idealised scenario in which the integration time is allowed to change in
different regions of the space, we show that geometric ergodicity can be
recovered for a much broader class of tail behaviours, leading to some
guidelines for the choice of this free parameter in practice.
