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 $Oleft(frac{d^{1/4}}{ varepsilon^{1/2}} right)$ 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 $O left(frac{d^{1/4}}{varepsilon^{1/2}} + frac{d^{1/2}}{varepsilon^{1/(alpha - 1)}} right)$.
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