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

• 成果类型: Article
• 署名作者: Morrow, GJ; Zhang, Y
• 署名单位: University of Colorado System; University of Colorado at Colorado Springs
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051605000000241
• 发表日期: 2005
• 页码: 1832-1886
• 关键词: brownian intersection exponents; plane exponents; VALUES

摘要

Let L-n denote the lowest crossing of a square 2n x 2n 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 Smimov 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 -> infinity: 1. E(vertical bar L(n)vertical bar(tau)) = n(4 tau/3+o(1)). 2. E(vertical bar F(n)vertical bar(tau)) = n(7 tau/4+o(1)). 3. E (vertical bar Q(n)vertical bar(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.