On the density of 2D critical percolation gaskets and anchored clusters

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number45
JournalLetters in Mathematical Physics
Issue number2
StatePublished - Apr 2024


  • 60J67
  • 81T27
  • 82B27
  • 82B43
  • Connection probabilities
  • Continuum scaling limit
  • Critical percolation
  • Primary 60K35
  • Secondary 82B31
  • Upper half-plane

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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