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The expected number of critical percolation clusters intersecting a line segment. (English) Zbl 1336.60196
Summary: 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). J. Cardy [“Conformal invariance and percolation”, Preprint, arXiv:math-ph/0103018] (see also [R. Yu, H. Saleur and S. Haas, “Entanglement entropy in the two-dimensional random transverse field Ising model”, Phys. Rev. B (3) 77, No. 14, Article ID 140402, 4 p. (2008; doi:10.1103/PhysRevB.77.140402)]) derived heuristically that \(E_{\mathbb H}(n) = An + \frac{\sqrt {3}} {4\pi }\log (n) + o(\log (n))\), where \(A\) is some constant. Recently, I. A. Kovács, F. Iglói and J. Cardy derived in [“Corner contribution to percolation cluster numbers”, Phys. Rev. B (3) 86, No. 21, Article ID 214203, 6 p. (2012; doi:10.1103/PhysRevB.86.214203)] 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 \(E_{\mathbb C}(n)\).
MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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