The scaling limit of critical hypercube percolation

We study the connected components in critical percolation on the Hamming hypercube $\{0,1{\}}^m$. We show that their sizes rescaled by $2^{-2m\mathrm{/}3}$ converge in distribution, and that, considered as metric measure spaces with the graph distance rescaled by $2^{-m\mathrm{/}3}$ and the uniform measure, they converge in distribution with respect to the Gromov-Hausdorff-Prokhorov topology. The two corresponding limits are as in critical Erdős-Rényi graphs.