Convergence of Random Batch Method with replacement for interacting particle systems

The paper proves that RBM‑r, run at time (N/p)t, converges to IPS at time t with rate sup_{0≤t≤T} W2(ρ̃^{(1)}_{(N/p)t}, ρ^{(1)}_t) ≤ C√(1+T)(N/p)^{3/4}κ^{1/4} under Assumption 2.1 (λ>2L, V strongly convex, K bounded Lipschitz) and common initial data, via a coupling/time‑change argument using an intermediate IPS’ process . The candidate solution reaches the same stated bound (and even sketches a stronger intermediate scaling O(√(κ/θ)) with θ=N/p), by a different approach based on unbiased generator perturbations, Dynkin’s formula, martingale (BDG-type) control, and a smoothing step. While the model’s outline has a few imprecisions (e.g., independence vs. martingale differences across windows, and the “weak-to-W2” smoothing step needing justification), its core steps align with standard techniques and recover the paper’s bound after a benign degradation √(κ/θ) ≤ θ^{3/4}κ^{1/4}. Hence both are correct, with substantially different proof strategies.