  We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order
Langevin dynamics for sampling from distributions with log-concave and smooth
densities. The higher-order dynamics allow for more flexible discretization
schemes, and we develop a specific method that combines splitting with more
accurate integration. For a broad class of $d$-dimensional distributions
arising from generalized linear models, we prove that the resulting third-order
algorithm produces samples from a distribution that is at most $\varepsilon >
0$ in Wasserstein distance from the target distribution in
$O\left(\frac{d^{1/4}}{ \varepsilon^{1/2}} \right)$ steps. This result requires
only Lipschitz conditions on the gradient. For general strongly convex
potentials with $\alpha$-th order smoothness, we prove that the mixing time
scales as $O \left(\frac{d^{1/4}}{\varepsilon^{1/2}} +
\frac{d^{1/2}}{\varepsilon^{1/(\alpha - 1)}} \right)$.
