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Finite-Size Left-Passage Probability in Percolation

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Abstract

We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. In terms of clusters, this corresponds to the one-arm probability. Our calculation is based on the q-deformed Knizhnik–Zamolodchikov approach, and the results are expressed in terms of symplectic characters. In the large size limit, we recover the scaling behaviour predicted by Schramm’s left-passage formula. We also derive a general relation between the left-passage probability in the Fortuin–Kasteleyn cluster model and the magnetisation profile in the open XXZ chain with diagonal, complex boundary terms.

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Notes

  1. The term ‘boundary edge’ is used throughout this paper to mean the edge closest to the left boundary along the given row.

  2. Note that this proof is only valid for odd L; for even L the statement must also be proved for L=2. We omit this part of the proof as we are only interested in odd system sizes.

  3. The definition of the transfer matrix implies that the components of the eigenvector, and thus the passage probabilities, are only functions of \(z_{i}^{2}\) and do not depend on any odd powers of z i . This fact is crucial to these proofs.

  4. We could also define \(\widehat{Y}_{j}\) in a similar fashion, but this is simply related to Y j by the transformation z k →1/z k , ∀k.

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Acknowledgements

The authors thank J. de Gier, C. Hagendorf, A. Mays, V. Terras, and P. Zinn-Justin for fruitful discussions. This work was supported by the European Research Council (grant CONFRA 228046).

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  1. Section de Mathématiques, University of Geneva, Rue du Lièvre 2-4, Geneva, Switzerland

    Yacine Ikhlef & Anita K. Ponsaing

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  1. Yacine Ikhlef
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  2. Anita K. Ponsaing
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Correspondence to Anita K. Ponsaing.

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Ikhlef, Y., Ponsaing, A.K. Finite-Size Left-Passage Probability in Percolation. J Stat Phys 149, 10–36 (2012). https://doi.org/10.1007/s10955-012-0573-z

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