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We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-orderLangevin dynamics for sampling from distributions with log-concave and smoothdensities. The higher-order dynamics allow for more flexible discretizationschemes, and we develop a specific method that combines splitting with moreaccurate integration. For a broad class of $d$-dimensional distributionsarising from generalized linear models, we prove that the resulting third-orderalgorithm 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 requiresonly Lipschitz conditions on the gradient. For general strongly convexpotentials with $\alpha$-th order smoothness, we prove that the mixing timescales as $O \left(\frac{d^{1/4}}{\varepsilon^{1/2}} +\frac{d^{1/2}}{\varepsilon^{1/(\alpha - 1)}} \right)$.

High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm