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

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

2005-08-24

arxiv.org/abs/math/0508459v1

Annals of Applied Probability 2005, Vol. 15, No. 3, 1832-1886

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051605000000241 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051605000000241

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.

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