The distribution of the minimum height among pivotal sites in critical two-dimensional percolation
| dc.creator | Morrow, Gregory J. | |
| dc.creator | Zhang, Yu | |
| dc.date | 2005-07-27 | |
| dc.date.accessioned | 2026-07-07T05:22:03Z | |
| dc.date.available | 2026-07-07T05:22:03Z | |
| dc.description | Let L_n denote the lowest crossing of the 2n \times 2n square box B(n) centered at the origin for critical site percolation on Z^2 or critical site percolation on the triangular lattice imbedded in Z^2, and denote by Q_n the set of pivotal sites along this crossing. On the event that a pivotal site exists, denote the minimum height that a pivotal site attains above the bottom of B(n) by M_n:= min{m:(x,-n+m)\in Q_n for some -n\le x\le n}. Else, define M_n = 2n. We prove that P(M_n < m) \asymp m/n, uniformly for 1\le m\le n. This relation extends Theorem 1 of van den Berg and Jarai (2003) who handle the corresponding distribution for the lowest crossing in a slightly different context. As a corollary we establish the asymptotic distribution of the minimum height of the set of cut points of a certain chordal SLE_6 in the unit square of C. | |
| dc.description | 20 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0507566 | |
| dc.identifier | http://arxiv.org/abs/math/0507566 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75915 | |
| dc.subject | Probability | |
| dc.subject | 60K35; 82B43 | |
| dc.title | The distribution of the minimum height among pivotal sites in critical two-dimensional percolation | |
| dc.type | text |