The expected number of critical percolation clusters intersecting a line segment

Abstract

We study critical percolation on a regular planar lattice. Let EG(n) be the expected number of open clusters intersecting or hitting the line segment [0; n]. (For the subscript G we either take H, when we restrict to the upper halfplane, or C, when we consider the full lattice). Cardy [2] (see also Yu, Saleur and Haas [11]) derived heuristically that EH(n) = An + p 3 4 log(n) + o(log(n)), where A is some constant. Recently Kovács, Iglói and Cardy derived in [5] heuristically (as a special case of a more general formula) that a similar result holds for EC(n) with the constant p 3 4 replaced by 5 p 3 32 . In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of EH(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of EC(n).