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 (2001) (see also Yu, Saleur and Haas (2008)) 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 (2012) derived 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 $E_{\mathbb{C}}(n)$.

Final version, appeared in Elect.Comm.Probab. 21 (2016)

References in corpus (3)

• Entanglement Entropy in the Two-Dimensional Random Transverse Field Ising Model
• Percolation Crossing Formulas and Conformal Field Theory
• Corner contribution to percolation cluster numbers

Cited by in corpus (1)

• Cluster tomography in percolation