Abstract
We prove a formula, first obtained by Kleban, Simmons and Ziff using conformal field theory methods, for the (renormalized) density of a critical percolation cluster in the upper half-plane “anchored” to a point on the real line. The proof is inspired by the method of images. We also show that more general bulk-boundary connection probabilities have well-defined, scale-covariant scaling limits and prove a formula for the scaling limit of the (renormalized) density of the critical percolation gasket in any domain conformally equivalent to the unit disk.
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Acknowledgements
The author thanks Peter Kleban and Robert Ziff for an interesting correspondence and for comments on a draft of the paper, as well as two anonymous referees for their useful suggestions.
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Camia, F. On the density of 2D critical percolation gaskets and anchored clusters. Lett Math Phys 114, 45 (2024). https://doi.org/10.1007/s11005-024-01793-0
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DOI: https://doi.org/10.1007/s11005-024-01793-0