In high-dimensional critical percolation, the rescaled cluster measure converges to the super-Brownian occupation measure, the rescaled cluster converges to its range in Hausdorff distance, and the one-arm probability has exact r^{-2} asymptotics.
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Range convergence and sharp one-arm asymptotics for the critical percolation cluster in high dimensions
In high-dimensional critical percolation, the rescaled cluster measure converges to the super-Brownian occupation measure, the rescaled cluster converges to its range in Hausdorff distance, and the one-arm probability has exact r^{-2} asymptotics.