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Inhomogeneous percolation on the hierarchical configuration model with a heavy-tailed degree distribution

David Clancy

Abstract

We consider inhomogeneous percolation on a hierarchical configuration model with a heavy-tailed degree distribution. This graph is the configuration model where all the half-edges are colored either black or white, and edges are formed by uniformly matching edges of the same color. When only the white half-edges are paired, we provide sufficient conditions for the size and total number of incident black half-edges of the connected components to converge in an $\ell^2$-sense. The limiting vector is described by an $\mathbb{R}^2$-valued thinned Lévy process. We also establish an $\ell^2$-limit for the number of vertices in connected components when a critical proportion of the black edges are included. A key part of our analysis is establishing a Feller-type property for the multiplicative coalescent with mass and weight recently studied in (Dhara et. al 2017, Dhara et. al 2020).

Inhomogeneous percolation on the hierarchical configuration model with a heavy-tailed degree distribution

Abstract

We consider inhomogeneous percolation on a hierarchical configuration model with a heavy-tailed degree distribution. This graph is the configuration model where all the half-edges are colored either black or white, and edges are formed by uniformly matching edges of the same color. When only the white half-edges are paired, we provide sufficient conditions for the size and total number of incident black half-edges of the connected components to converge in an -sense. The limiting vector is described by an -valued thinned Lévy process. We also establish an -limit for the number of vertices in connected components when a critical proportion of the black edges are included. A key part of our analysis is establishing a Feller-type property for the multiplicative coalescent with mass and weight recently studied in (Dhara et. al 2017, Dhara et. al 2020).
Paper Structure (33 sections, 39 theorems, 233 equations, 3 algorithms)

This paper contains 33 sections, 39 theorems, 233 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Description of Model
  3. Assumptions and Main Results
  4. Discussion and Overview
  5. Connection with the HCM in vdHvLS.18
  6. Overview of The Article
  7. The Multiplicative Coalescent with Mass and Weight
  8. Basic Lemmas and Notations
  9. Graphical Couplings
  10. Sufficient Condition for Theorem \ref{['thm:MC']}
  11. Proof of Theorem \ref{['thm:MC']}
  12. Approximate MCMW
  13. Exploration of the HCM
  14. Exploration of the White Configuration Model
  15. Incorporating the Black Edges
  16. ...and 18 more sections

Key Result

Theorem 1.1

Suppose that ${\bf x}^n,{\bf y}^n\in \ell^2_+$ and ${\bf x}^n\to {\bf x}$ and ${\bf y}^n\to {\bf y}$ in $\ell^2$. If each coordinate of ${\bf y}$ is strictly positive, then for any fixed $t\ge 0$

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.3: Dhara et. al.DvdHvLS.20
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.2: van der Hofstad et. al. vdHvLS.18
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Lemma 3.4
  • ...and 53 more