textabstractWe study critical percolation on a regular planar lattice. Let EG(n) be the expected number of open clusters intersecting or hitting the line segment [0, n]. (For the subscript G we either take ℍ(when we restrict to the upper halfplane, or ℂ, when we consider the full lattice). Cardy [2] (see also Yu, Saleur and Haas [11]) derived heuristically that Eℍ((n) =An +√3/4π 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 Eℂ(n) with the constant √3/4π replaced by 5√3/32π. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of Eℍ(n) above, and a rigorous upper bound for the prefactor of the logarithm in the formula of Eℂ(n).
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机译：我们研究了规则平面晶格上的临界渗滤。令EG（n）是相交或到达线段[0，n]的开放簇的预期数目。 （对于下标G，我们要么取ℍ（当我们限制为上半平面时，要么取ℂ，当我们考虑全晶格时。）Cardy [2]（另请参见Yu，Saleur和Haas [11]）启发式地得出Eℍ（ （n）= An +√3/4πlog（n）+ o（log（n）），其中A是一个常数。最近Kovács，Iglói和Cardy启发式地从[5]中推导（作为更一般情况的特例）公式），对于Eℂ（n）具有类似的结果，常数√3/4π被5√3/32π取代。对于三角形格上的位置渗漏，本文给出了Eℍ（n）公式的严格证明n），以及Eℂ（n）公式中对数的因式的严格上限。
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