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The random batch method provides an efficient algorithm for computingstatistical properties of a canonical ensemble of interacting particles. Inthis work, we study the error estimates of the fully discrete random batchmethod, especially in terms of approximating the invariant distribution. Usinga triangle inequality framework, we show that the long-time error of the methodis $O(\sqrt{\tau} + e^{-\lambda t})$, where $\tau$ is the time step and$\lambda$ is the convergence rate which does not depend on the time step $\tau$or the number of particles $N$. Our results also apply to the McKean-Vlasovprocess, which is the mean-field limit of the interacting particle system asthe number of particles $N\rightarrow\infty$.

Error Analysis of Time-Discrete Random Batch Method for Interacting Particle Systems and Associated Mean-Field Limits