2 7 Ju l 2 00 5 The distribution of the minimum height among pivotal sites in critical two - dimensional percolation ∗

Let Ln denote the lowest crossing of the 2n × 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 Qn 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 Mn := min{m ≥ 0 : (x, −n + m) ∈ Qn for some − n ≤ x ≤ n} Else, define Mn = 2n. We prove that P (Mn < m) ≍ m/n, uniformly for 1 ≤ m ≤ n. This relation extends Theorem 1 of van den Berg and Jarai [2] 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 SLE6 in the unit square of C.