Critical percolation: the expected number of clusters in a rectangle

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results a...

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Veröffentlicht in:	Probability theory and related fields 2011-12, Vol.151 (3-4), p.735-756
Hauptverfasser:	Hongler, Clément, Smirnov, Stanislav
Format:	Artikel
Sprache:	eng
Schlagworte:	Clusters Economics Exact sciences and technology Finance General topics Insurance Invariance Invariants Lattices Management Mathematical analysis Mathematical and Computational Biology Mathematical and Computational Physics Mathematics Mathematics and Statistics Operations Research/Decision Theory Percolation Probability Probability and statistics Probability theory Probability Theory and Stochastic Processes Proving Quantitative Finance Random variables Rectangles Sciences and techniques of general use Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications) Statistics for Business Studies Theoretical
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Zusammenfassung:	We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and might provide a new approach to establishing conformal invariance of percolation.
ISSN:	0178-8051 1432-2064
DOI:	10.1007/s00440-010-0313-8