Convergence rate in a martingale CLT for percolation clusters
渗流簇中鞅中心极限定理的收敛速度

Consider bond percolation on Zd with parameter p. Let Kn be the number of open clusters in -n,n]d. We investigate the convergence rate in the martingaleCLT for Kn. Generally, the best convergence rate for classicalmartingale CLT is O(n-d/2), and our result is Pp((Kn-Ep(Kn))/(Varp(Kn))) ≤x =x∫-∞(1/(2π)) e(-y2)/2dy+o(n-d/2 +ε0) for all x, where ε0 is any constant real number in 0, d/2 . As far as we know, this is the first convergence rate in CLTs for percolation.