Finite-size left-passage probability in percolation

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. 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.

21 pages, 8 figures

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Cited by in corpus (3)

• Planar random-cluster model: fractal properties of the critical phase
• Schramm's formula for multiple loop-erased random walks
• On the logarithmic bipartite fidelity of the open XXZ spin chain at $\Delta =-1/2$