  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.
