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The scaling limit of critical hypercube percolation
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abstract
We study the connected components in critical percolation on the Hamming hypercube $\{0,1\}^m$. We show that their sizes rescaled by $2^{-2m/3}$ converge in distribution, and that, considered as metric measure spaces with the graph distance rescaled by $2^{-m/3}$ and the uniform measure, they converge in distribution with respect to the Gromov-Hausdorff-Prokhorov topology. The two corresponding limits are as in critical Erd\H{o}s-R\'enyi graphs.
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Super-Brownian limits and the $k$-point function for high-dimensional percolation
High-dimensional critical percolation clusters rescale to super-Brownian excursion, verifying the 1984 Aizenman–Newman k-point conjecture under lace-expansion hypotheses.
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