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 Kov\'{a}cs, Igl\'{o}i 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 $E_{\mathbb{C}}(n)$.
Highlights
1.1 Background and statement of the main resultConsider critical bond percolation on Z2
Iglói and Cardy [5] studied the expected number of clusters which intersect the boundary of a polygon
The prefactor of this term is lattice dependent. Their main interest is in the first correction term. Their motivation came from relations with entanglement entropy in a diluted quantum Ising model
Summary
The leading order is the size n of the boundary The prefactor of this term is lattice dependent. A special case of their result is that of a line segment (treated in Section F of their paper) In their setup the l√ine segment was placed in the full plane and they claim that the prefactor is equal to they refer to an earlier obtained result by. Cardy in [2] (see Yu, Saleur and Haas [11]) where the line segment was placed on t√he boundary of the half-plane. In the latter case the claim is that the prefactor equals. In the case of the half-plane we could obtain a rigorous proof for the earlier mentioned logarithmic correction term. Our main contribution is a rigorous proof of the following: Theorem 1.1. (a)
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