  We study the geometric ergodicity and the long time behavior of the Random
Batch Method for interacting particle systems, which exhibits superior
numerical performance in recent large-scale scientific computing experiments.
We show that for both the interacting particle system (IPS) and the random
batch interacting particle system (RB-IPS), the distribution laws converge to
their respective invariant distributions exponentially, and the convergence
rate does not depend on the number of particles $N$, the time step $\tau$ for
batch divisions or the batch size $p$. Moreover, the Wasserstein distance
between the invariant distributions of the IPS and the RB-IPS is bounded by
$O(\sqrt{\tau})$, showing that the RB-IPS can be used to sample the invariant
distribution of the IPS accurately with greatly reduced computational cost.
