Electronic Communications in Probability

The expected number of critical percolation clusters intersecting a line segment

J. van den Berg and R.P. Conijn

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

critical percolation number of clusters logarithmic correction term

Creative Commons Attribution 4.0 International License.


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

Export citation


  • [1] J.L. Cardy, Critical percolation in finite geometries, Journal of Physics A: Mathematical and General 25 (1992), no. 4, L201–L206.
  • [2] J.L. Cardy, Lectures on conformal invariance and percolation, arXiv:0103018 (2001).
  • [3] Julien Dubédat, Excursion decompositions for SLE and Watts’ crossing formula, Probab. Theory Related Fields 134 (2006), no. 3, 453–488.
  • [4] Clément Hongler and Stanislav Smirnov, Critical percolation: the expected number of clusters in a rectangle, Probability theory and related fields 151 (2011), no. 3-4, 735–756.
  • [5] István A. Kovács, Ferenc Iglói, and John Cardy, Corner contribution to percolation cluster numbers, Phys. Rev. B 86 (2012), 1–6.
  • [6] P. Nolin, Near-critical percolation in two dimensions, Electron. J. Probab. 13 (2008), no. 55, 1562–1623.
  • [7] Scott Sheffield and David B. Wilson, Schramm’s proof of Watts’ formula, The Annals of Probability 39 (2011), no. 5, 1844–1863.
  • [8] Jacob J.H. Simmons, Peter Kleban, and Robert M. Ziff, Percolation crossing formulae and conformal field theory, Journal of Physics A: Mathematical and Theoretical 40 (2007), no. 31, F771.
  • [9] Stanislav Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244.
  • [10] G.M.T. Watts, A crossing probability for critical percolation in two dimensions, Journal of Physics A: Mathematical and General 29 (1996), no. 14, L363.
  • [11] Rong Yu, Hubert Saleur, and Stephan Haas, Entanglement entropy in the two-dimensional random transverse field Ising model, Physical Review B 77 (2008), no. 14, 140402.