Convergence of $k$-point functions in high dimensional percolation

Consider critical Bernoulli percolation on ${\mathbb{Z}}^d$ for $d$ large; let $y_0,\dots ,y_{k-1}$ be $k$ distinct points in ${\mathbb{R}}^d$. We prove that the probability that $\{\lfloor n{y}_i\rfloor {\}}_{i=0}^{k-1}$ all lie in the same open cluster, rescaled by an appropriate power of $n$, converges as $n\to \mathrm{\infty }$ to an explicit constant. This confirms a conjecture of Aizenman and Newman.