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Diameter and mixing time of the giant component in the percolated hypercube

Michael Anastos, Sahar Diskin, Lyuben Lichev, Maksim Zhukovskii

TL;DR

This work resolves longstanding questions on the giant component of the percolated hypercube $Q^d_p$ with $p=c/d$ ($c>1$) by proving the giant’s diameter is $Θ(d)$ and the lazy-walk mixing time on it is $Θ(d^2)$. The authors develop a tight large-deviation framework for the giant’s size, built from a structural description of the residue after sprinkling, a quantitative expansion theory, and a stability principle against thinning, enabling optimal expansion bounds. A novel reverse-sprinkling approach, together with detailed decomposition and spread analyses, underpins both the lower-tail control and the expansion results, which feed into the mixing-time bound via a FR07a-type conductance argument. The techniques bypass reliance on the hypercube’s product structure and yield tools potentially applicable to other sparse high-dimensional percolation models, advancing our understanding of the interplay between connectivity, expansion, and random-walk mixing in giant components.

Abstract

We consider bond percolation on the $d$-dimensional binary hypercube with $p=c/d$ for fixed $c>1$. We prove that the typical diameter of the giant component $L_1$ is of order $Θ(d)$, and the typical mixing time of the lazy random walk on $L_1$ is of order $Θ(d^2)$. This resolves long-standing open problems of Bollobás, Kohayakawa and Łuczak from 1994, and of Benjamini and Mossel from 2003. A key component in our approach is a new tight large deviation estimate on the number of vertices in $L_1$ whose proof includes several novel ingredients: a structural description of the residue outside the giant component after sprinkling, a tight quantitative estimate on the spread of the giant in the hypercube, and a stability principle which rules out the disintegration of large connected sets under thinning. This toolkit further allows us to obtain optimal bounds on the expansion in $L_1$.

Diameter and mixing time of the giant component in the percolated hypercube

TL;DR

This work resolves longstanding questions on the giant component of the percolated hypercube with () by proving the giant’s diameter is and the lazy-walk mixing time on it is . The authors develop a tight large-deviation framework for the giant’s size, built from a structural description of the residue after sprinkling, a quantitative expansion theory, and a stability principle against thinning, enabling optimal expansion bounds. A novel reverse-sprinkling approach, together with detailed decomposition and spread analyses, underpins both the lower-tail control and the expansion results, which feed into the mixing-time bound via a FR07a-type conductance argument. The techniques bypass reliance on the hypercube’s product structure and yield tools potentially applicable to other sparse high-dimensional percolation models, advancing our understanding of the interplay between connectivity, expansion, and random-walk mixing in giant components.

Abstract

We consider bond percolation on the -dimensional binary hypercube with for fixed . We prove that the typical diameter of the giant component is of order , and the typical mixing time of the lazy random walk on is of order . This resolves long-standing open problems of Bollobás, Kohayakawa and Łuczak from 1994, and of Benjamini and Mossel from 2003. A key component in our approach is a new tight large deviation estimate on the number of vertices in whose proof includes several novel ingredients: a structural description of the residue outside the giant component after sprinkling, a tight quantitative estimate on the spread of the giant in the hypercube, and a stability principle which rules out the disintegration of large connected sets under thinning. This toolkit further allows us to obtain optimal bounds on the expansion in .
Paper Structure (20 sections, 35 theorems, 137 equations, 1 figure)

This paper contains 20 sections, 35 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the proofs
  3. Preliminaries
  4. Notation
  5. Concentration inequalities
  6. Auxiliary graph-theoretic results
  7. Auxiliary results on percolation
  8. Branching processes and the BFS algorithm
  9. Upper tail
  10. Lower tail
  11. Vertices in large components
  12. Typical properties of G1 and G2
  13. Proof of the lower tail bound of Theorem \ref{['th: large tail deviation']}
  14. Mixing time
  15. Expansion properties of the giant
  16. ...and 5 more sections

Key Result

Theorem 1

Fix $c > 1$ and let $p=p(d)=c/d$. Then whpWith high probability, that is, with probability tending to $1$ as $d\to \infty$. the giant component $L_1$ in $Q^d_p$ satisfies each of the following properties.

Figures (1)

  • Figure 1: Illustration of some of the sets from the last proof. The family $\mathcal{M}_{10}$ is represented by a rectangle with a thick black border, and the family $\mathcal{M}_2^-$ is represented by a light-blue shape with a dashed border. The family $\mathcal{C}$ of components in $\mathcal{M}_2^-$ which are not part of $L_1'$ and the set of vertices in $S_{\mathcal{C}}$ attached to it through edges of $Q^d_{p_2}$ are both represented in red. The family $\mathcal{S}$ of components which are in $\mathcal{M}_{10}$ but do not intersect $\mathcal{M}_2^-$ is represented in yellow. $L_1'$ consists of the light-blue and the white region.

Theorems & Definitions (75)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • Corollary 3.6
  • ...and 65 more