The expected number of critical percolation clusters intersecting a line segment

TL;DR

This paper rigorously proves the asymptotic behavior of the expected number of critical percolation clusters intersecting a line segment in the upper half-plane and provides bounds for the full lattice case, confirming heuristic predictions.

Contribution

It provides a rigorous proof for the expected number of clusters in the half-plane and bounds for the full lattice, advancing understanding of critical percolation cluster behavior.

Findings

Rigorous proof of the asymptotic formula for $E_{ ext{half-plane}}(n)$.

Upper bounds for the logarithmic term in $E_{ ext{full lattice}}(n)$.

Confirmation of heuristic constants in critical percolation theory.

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 $E_{\mathbb{C}}(n)$.