On the number of pioneers in high-dimensional critical percolation

We prove that for critical percolation in high dimension the expected number of pioneers on any set containing the origin is of the same order as the probability that a simple random walk starting from the origin exits the set. Our proof works in all dimensions $d>6$, under the usual assumption that the two-point function is of the same order as the Green's function. As an application we obtain a sharp uniform quantitative bound on the probability that the cluster of the origin intersects any given ball.