Microscopic derivation of the Keller-Segel equation in the sub-critical regime

We derive the two-dimensional Keller-Segel equation from a stochastic system of $N$ interacting particles in the case of sub-critical chemosensitivity $\chi < 8 \pi$. The Coulomb interaction force is regularised with a cutoff of size $N^{- \alpha}$, with arbitrary $\alpha \in (0, 1 / 2)$. In particular we obtain a quantitative result for the maximal distance between the real and mean-field $N$-particle trajectories.