Exact results for the universal area distribution of clusters in percolation, Ising and Potts models

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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.