Slightly subcritical hypercube percolation

TL;DR

This paper analyzes the structure and properties of the largest clusters in slightly subcritical bond percolation on hypercubes, providing precise estimates for their size, diameter, and mixing time, with implications for various high-dimensional graphs.

Contribution

It establishes detailed asymptotic behaviors of cluster sizes, diameters, and mixing times in the slightly subcritical regime, extending results to various high-dimensional graph classes.

Findings

Largest component size: Θ(ε_m^{-2} log(ε_m^3 2^m))

Maximum diameter: (1+o(1)) ε_m^{-1} log(ε_m^3 2^m)

Maximum mixing time: Θ(ε_m^{-3} log^2(ε_m^3 2^m))

Abstract

We study bond percolation on the hypercube $\{0,1\}^m$ in the slightly subcritical regime where $p = p_c (1-\varepsilon_m)$ and $\varepsilon_m = o(1)$ but $\varepsilon_m \gg 2^{-m/3}$ and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality $\Theta\left(\varepsilon_m^{-2} \log(\varepsilon_m^3 2^m)\right)$, that the maximal diameter of all clusters is $(1+o(1)) \varepsilon_m^{-1} \log(\varepsilon_m^3 2^m)$, and that the maximal mixing time of all clusters is $\Theta\left(\varepsilon_m^{-3} \log^2(\varepsilon_m^3 2^m)\right)$. 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.