The second largest component in the supercritical 2D Hamming graph

TL;DR

This paper investigates the size of the second largest component in a supercritical random subgraph of the 2D Hamming graph, revealing it is approximately proportional to ^{-2}, indicating the emergence of the dominant component.

Contribution

It establishes the size of the second largest component in the supercritical phase of the 2D Hamming graph, suggesting a possible discrete duality principle.

Findings

Second largest component size is close to ^{-2} in the supercritical regime.

Results support the emergence of a dominant component after percolation.

Indicates a potential duality between supercritical and subcritical regimes.

Abstract

The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices $(i,j)$, $1\leq i,j\leq n$, 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 $2(n-1)p=1+\epsilon$. Previous work by van der Hofstad and Luczak had shown that in the barely supercritical region $n^{-2/3}\ln^{1/3}n\ll \epsilon \ll 1$ the largest component has size $\sim 2\epsilon n$. Here we show that the second largest component has size close to $\epsilon^{-2}$, 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.