Capacity in high dimensional percolation

We introduce a notion of capacity for high dimensional critical percolation by showing that for any finite set $A$, the suitably rescaled probability that the cluster of $z$ intersects $A$ converges as $\|z\|\to\infty$. This can be viewed as a generalisation of the asymptotic of the two point function and we call the limit the p-capacity of $A$. We next show that the probability that the Incipient Infinite Cluster of $z$ intersects the set $A$ appropriately normalised is also of order the p-capacity of $A$ as $\|z\|\to\infty$. We conjecture that the p-capacity is of the same order as the $(d-4)$-Bessel-Riesz capacity and in support of this we estimate the p-capacity of balls. As a byproduct of our techniques we give a simpler proof of the one-arm exponent of Kozma and Nachmias for dimensions 8 and higher and as long as the two point function asymptotic holds. Our proofs make use of a new large deviations bound on the pioneers, that is the number of points on the boundary of a box which are part of the cluster of the origin restricted to this box.