Abstract
We study bond percolation on the hypercube {0,1}m in the slightly subcritical regime where p = pc(1 − εm) and εm = o(1) but εm ≫ 2−m/3 and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality , that the maximal diameter of all clusters is , and that the maximal mixing time of all clusters is .These results hold in different levels of generality, and in particular, some of the estimates hold for various classes of graphs such as high‐dimensional tori, expanders of high degree and girth, products of complete graphs, and infinite lattices in high dimensions.
Highlights
AND MAIN RESULTSThe hypercube Qm is the graph with vertex set {0, 1}m such that any two vertices of Hamming distance 1 form an edge
We consider bond percolation on it, that is, the random subgraph of Qm obtained by independently removing each edge with probability 1 − p ∈ [0, 1] and retaining it otherwise
Hypercube percolation was introduced by Erdos and Spencer [13] and is compared there with the Erdos-Rényi random graph (ERRG) G(n, p), which is bond percolation on the complete graph Kn with percolation parameter p
Summary
The hypercube Qm is the graph with vertex set {0, 1}m such that any two vertices of Hamming distance 1 form an edge. We put p = c∕n for some constant c, and write j for the jth largest connected component of G(n, p) It holds that when c < 1 we have | 1| = Θ(log n) whp, while when c > 1 we have that | 1| = Θ(n) whp [12]. It has been shown so far that many of the features of the ERRG phase transition hold for hypercube percolation. To state these results we first need to discuss the percolation threshold probability. (see [18] for an elementary proof)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
