English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let L_n denote the lowest crossing of a square 2n\times2n box for criticalsite percolation on the triangular lattice imbedded in Z^2. Denote also by F_nthe pioneering sites extending below this crossing, and Q_n the pivotal siteson this crossing. Combining the recent results of Smirnov and Werner [Math.Res. Lett. 8 (2001) 729-744] on asymptotic probabilities of multiple arm pathsin both the plane and half-plane, Kesten's [Comm. Math. Phys. 109 (1987)109-156] method for showing that certain restricted multiple arm paths areprobabilistically equivalent to unrestricted ones, and our own second andhigher moment upper bounds, we obtain the following results. For each positiveinteger \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)}. Theseresults extend to higher moments a discrete analogue of the recent results ofLawler, Schramm and Werner [Math. Res. Lett. 8 (2001) 401-411] that thefrontier, pioneering points and cut points of planar Brownian motion haveHausdorff dimensions, respectively, 4/3, 7/4 and 3/4.

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