The sizes of the pioneering, lowest crossing and pivotal sites in critical percolation on the triangular lattice

Abstract

Let L_n denote the lowest crossing of a square 2n×2n box for critical site percolation on the triangular lattice imbedded in Z². 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 τ, as n→∞:

1. E(|L_n|^τ)=n^4τ/3+o(1).

2. E(|F_n|^τ)=n^7τ/4+o(1).

3. E(|Q_n|^τ)=n^3τ/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.