## Electronic Communications in Probability

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

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 28, 10 pp.

Dates
Received: 28 July 2015
Accepted: 15 March 2016
First available in Project Euclid: 18 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1458324257

Digital Object Identifier
doi:10.1214/16-ECP4452

Mathematical Reviews number (MathSciNet)
MR3485397

Zentralblatt MATH identifier
1336.60196

#### Citation

van den Berg, J.; Conijn, R.P. The expected number of critical percolation clusters intersecting a line segment. Electron. Commun. Probab. 21 (2016), paper no. 28, 10 pp. doi:10.1214/16-ECP4452. https://projecteuclid.org/euclid.ecp/1458324257

#### References

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