  We propose a fast potential splitting Markov Chain Monte Carlo method which
costs $O(1)$ time each step for sampling from equilibrium distributions (Gibbs
measures) corresponding to particle systems with singular interacting kernels.
We decompose the interacting potential into two parts, one is of long range but
is smooth, and the other one is of short range but may be singular. To displace
a particle, we first evolve a selected particle using the stochastic
differential equation (SDE) under the smooth part with the idea of random
batches, as commonly used in stochastic gradient Langevin dynamics. Then, we
use the short range part to do a Metropolis rejection. Different from the
classical Langevin dynamics, we only run the SDE dynamics with random batch for
a short duration of time so that the cost in the first step is $O(p)$, where
$p$ is the batch size. The cost of the rejection step is $O(1)$ since the
interaction used is of short range. We justify the proposed random-batch Monte
Carlo method, which combines the random batch and splitting strategies, both in
theory and with numerical experiments. While giving comparable results for
typical examples of the Dyson Brownian motion and Lennard-Jones fluids, our
method can save more time when compared to the classical Metropolis-Hastings
algorithm.
