The scaling limit of critical hypercube percolation

TL;DR

This paper investigates the scaling limits of connected components in critical hypercube percolation, demonstrating convergence in distribution to limits similar to those in critical Erdős-Rényi graphs, with implications for understanding phase transitions.

Contribution

It establishes the convergence of component sizes and metric measure spaces in critical hypercube percolation, extending known results from Erdős-Rényi graphs to high-dimensional hypercubes.

Findings

Component sizes rescaled by 2^{-2m/3} converge in distribution.

Metric measure spaces converge in Gromov-Hausdorff-Prokhorov topology.

Limits match those of critical Erdős-Rényi graphs.

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.