Finite-size left-passage probability in percolation

Details Finite-size left-passage probability in percolation Finite-size left-passage probability in percolation

2012-02-24

arxiv.org/abs/1202.5476v1

Physics

Mathematical Physics

21 pages, 8 figures

Scientific paper

We obtain an exact finite-size expression for the probability that a percolation hull will touch the boundary, on a strip of finite width. 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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