The expected number of critical percolation clusters intersecting a line segment

Abstract

We study critical percolation on a regular planar lattice. Let $E_G(n)$ be the expected number of open clusters intersecting or hitting the line segment $[0,n]$. (For the subscript $G$ we either take $\mathbb{H} $, when we restrict to the upper halfplane, or $\mathbb{C} $, when we consider the full lattice).

Cardy [2] (see also Yu, Saleur and Haas [11]) derived heuristically that $E_{\mathbb{H} }(n) = An + \frac{\sqrt {3}} {4\pi }\log (n) + o(\log (n))$, 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 $E_{\mathbb{C} }(n)$ with the constant $\frac{\sqrt {3}} {4\pi }$ replaced by $\frac{5\sqrt {3}} {32\pi }$.

In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of $E_{\mathbb{H} }(n)$ above, and a rigorous upper bound for the prefactor of the logarithm in the formula of $E_{\mathbb{C} }(n)$.