On the rate of normal approximation for Poisson continuum percolation

It is known that the number of points in the largest cluster of a percolating Poisson process restricted to a large finite box is asymptotically normal. In this note, we establish a rate of convergence for the statement. As each point in the largest cluster is determined by points as far as the diameter of the box, known results in the literature of normal approximation for Poisson functionals cannot be directly applied. To disentangle the long-range dependence of the largest cluster, we use the fact that the second largest cluster has comparatively shorter range of dependence to restrict the range of dependence, apply a recently established result in Chen, Röllin and Xia (2021) to obtain a Berry-Esseen type bound for the normal approximation of the number of points belonging to clusters that have a restricted range of dependence, and then estimate the gap between this quantity and the number of points in the largest cluster.