Exact results for the universal area distribution of clusters in percolation, Ising and Potts models
Abstract
At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to 1/A, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that C = 1/(8 sqrt(3) pi) = 0.022972037.... We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin-Kasteleyn clusters of the Q=2, 3 and 4-state Potts models.
0
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.