  Let L_n denote the lowest crossing of a square 2n\times2n box for critical
site percolation on the triangular lattice imbedded in Z^2. Denote also by F_n
the pioneering sites extending below this crossing, and Q_n the pivotal sites
on this crossing. Combining the recent results of Smirnov and Werner [Math.
Res. Lett. 8 (2001) 729-744] on asymptotic probabilities of multiple arm paths
in both the plane and half-plane, Kesten's [Comm. Math. Phys. 109 (1987)
109-156] method for showing that certain restricted multiple arm paths are
probabilistically equivalent to unrestricted ones, and our own second and
higher moment upper bounds, we obtain the following results. For each positive
integer \tau, as n\to\infty: 1. E(|L_n|^{\tau})=n^{4\tau/3+o(1)}. 2.
E(|F_n|^{\tau})=n^{7\tau/4+o(1)}. 3. E(|Q_n|^{\tau})=n^{3\tau/4+o(1)}. These
results extend to higher moments a discrete analogue of the recent results of
Lawler, Schramm and Werner [Math. Res. Lett. 8 (2001) 401-411] that the
frontier, pioneering points and cut points of planar Brownian motion have
Hausdorff dimensions, respectively, 4/3, 7/4 and 3/4.
