  The random batch method provides an efficient algorithm for computing
statistical properties of a canonical ensemble of interacting particles. In
this work, we study the error estimates of the fully discrete random batch
method, especially in terms of approximating the invariant distribution. Using
a triangle inequality framework, we show that the long-time error of the method
is $O(\sqrt{\tau} + e^{-\lambda t})$, 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-Vlasov
process, which is the mean-field limit of the interacting particle system as
the number of particles $N\rightarrow\infty$.

Probability;Numerical Analysis;Numerical Analysis