10 research outputs found
The second largest component in the supercritical 2D Hamming graph
The 2-dimensional Hamming graph H(2,n) consists of the vertices
, , two vertices being adjacent when they share a
common coordinate. We examine random subgraphs of H(2,n) in percolation with
edge probability , so that the average degree . Previous
work by van der Hofstad and Luczak had shown that in the barely supercritical
region the largest component has size
. Here we show that the second largest component has size
close to , so that the dominant component has emerged. This
result also suggests that a {\it discrete duality principle} might hold,
whereby, after removing the largest connected component in the supercritical
regime, the remaining random subgraphs behave as in the subcritical regime.Comment: 9 pages, revised versio
Expansion of percolation critical points for Hamming graphs
The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let be the degree and be the number of vertices of H(d, n). Let be the critical point for bond percolation on H(d, n). We show that, for fixed and ,
which extends the asymptotics found in [10] by one order. The term is the width of the critical window. For we have , and so the above formula represents the full asymptotic expansion of . In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for . The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös–Rényi random grap
The second largest component in the supercritical 2D Hamming graph
The 2-dimensional Hamming graph H(2,n) consists of the n2 vertices (i,j), 1≤ i,j≤ n, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability p, so that the average degree 2(n-1)p=1+ε. Previous work [5] had shown that in the barely supercritical region n-2/3 ln1/3n << ε << 1 the largest component has size ~ 2εn. Here we show that the second largest component has size close to ε-2, so that the dominant component has emerged