Hypercube percolation

TL;DR

This paper analyzes bond percolation on the hypercube near the critical probability, establishing precise component size behavior and resolving a conjecture about the phase transition in this high-dimensional setting.

Contribution

It proves that for percolation probabilities slightly above the critical point, the largest component size scales linearly with epsilon and the hypercube size, confirming a conjecture.

Findings

Largest component size is approximately 2*epsilon*2^m

Second largest component is negligible compared to the largest

Results hold with high probability for epsilon_m=o(1) and epsilon_m>>2^{-m/3}

Abstract

We study bond percolation on the Hamming hypercube {0,1}^m around the critical probability p_c. It is known that if p=p_c(1+O(2^{-m/3})), then with high probability the largest connected component C_1 is of size Theta(2^{2m/3}) and that this quantity is non-concentrated. Here we show that for any sequence eps_m such that eps_m=o(1) but eps_m >> 2^{-m/3} percolation on the hypercube at p_c(1+eps_m) has |C_1| = (2+o(1)) eps_m 2^m and |C_2| = o(eps_m 2^m) with high probability, where C_2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [17].