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The scaling limit of critical hypercube percolation

Arthur Blanc-Renaudie, Nicolas Broutin, Asaf Nachmias

Abstract

We study the connected components in critical percolation on the Hamming hypercube $\{0,1\}^m$. We show that their sizes rescaled by $2^{-2m/3}$ converge in distribution, and that, considered as metric measure spaces with the graph distance rescaled by $2^{-m/3}$ and the uniform measure, they converge in distribution with respect to the Gromov-Hausdorff-Prokhorov topology. The two corresponding limits are as in critical Erdős-Rényi graphs.

The scaling limit of critical hypercube percolation

Abstract

We study the connected components in critical percolation on the Hamming hypercube . We show that their sizes rescaled by converge in distribution, and that, considered as metric measure spaces with the graph distance rescaled by and the uniform measure, they converge in distribution with respect to the Gromov-Hausdorff-Prokhorov topology. The two corresponding limits are as in critical Erdős-Rényi graphs.
Paper Structure (23 sections, 43 theorems, 250 equations, 5 figures, 1 table)

This paper contains 23 sections, 43 theorems, 250 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Other underlying graphs
  5. Outline of the proof and organization
  6. Table of notations
  7. Preliminaries
  8. Topological notions
  9. Percolation
  10. Non-backtracking walk on the hypercube and percolation
  11. Convergence of the component multiplicative graph $G_\times$
  12. Subcritical estimates for $\sigma_2$ and $\sigma_3$: Proofs of Lemmas \ref{['lem:sigma2Concentration']} and \ref{['lem:sigma3Concentration']}
  13. Convergence of the susceptibility for multiplicative graph: Proof of \ref{['CVX']}
  14. Convergence of the sprinkled component graph $G_\mathfrak{C}$
  15. Bounding the $\ell^2$ distance between the edge probabilities
  16. ...and 8 more sections

Key Result

Theorem 1.1

Fix $\lambda \in \mathbb{R}$ and let $p_c=p_c(\lambda,m) \in(0,1)$ defined by def:pc. Consider the ordered connected components $(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ of percolation on the hypercube with edge probability $p_c$. Then where the convergence is with respect to $\ell^2$.

Figures (5)

  • Figure 1: A representation of the event $A_{x,y,z}$. The paths $\gamma_{v,z}$, $\gamma_{z,x}$, $\gamma_{z,y}$ are blue. The interior of the red curves represents $\mathcal{C}_v, \mathcal{C}_x,\mathcal{C}_y$ which do not contain $z$.
  • Figure 2: A representation from left to right of the events (a), (b), (c) of Lemma 4.15. There is a path between $u,v$ in $H_{p_c}$ crossing a small connected component $A$ of $H_{p_s}$.
  • Figure 3: The edge $(x,x')$ is removed when loop erasing $\Gamma'$ in two ways: on the left case (i) and on the right case (ii). The path $\gamma$ is represented in thick black/blue line. Red blobs represent some connected components of $H_{p_s}$. Blue paths have length at most $L$.
  • Figure 4: A path in $H_{p_c}$ with corresponding length $K=5$ in $G_{\tilde{\mathfrak{C}}}$ is represented. The red blobs represent connected components of $H_{p_s}$. Paths in $H_{p_s}$ are drawn in blue and edges of $H_{p_c}\backslash H_{p_s}$ are drawn in black.
  • Figure 5: The cycle event from the proof of Lemma \ref{['lem:GCompHaveNoSmallCycles']} is represented. The red blobs represent some connected components of $H_{p_s}$. Blue paths are in $H_{p_s}$. Black segment are edges in $H_{p_c}\backslash H_{p_s}$. The path between $x$ and $z'$ represented in red is in $H_{p_c}$.

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorems \ref{['main_thm']} and \ref{['thm:main_metric']} assuming \ref{['main_thm_general']}
  • Remark 1.3.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.2.1
  • Remark 2.2.2
  • Proposition 3.1
  • ...and 71 more