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The critical percolation window in growing random graphs

Joost Jorritsma, Pascal Maillard, Peter Mörters

TL;DR

We study critical percolation for sparse growing random graphs, introducing the γ-growing model with sequential vertex arrivals and probabilistic connections that mimic preferential attachment. The core method couples component explorations to a two-sided killed branching random walk, enabling precise control of barely subcritical and critical regimes. The main findings establish a critical window of width $(\log n)^{-2}$ with the largest component of order $\sqrt{n}/\log n$ and finite, universal susceptibility across the window, plus a secondary transition at the upper boundary. Together, these results delineate a universality class for growing graphs, distinct from mean-field and rank-one models, with a local limit described by the trace of a killed BRW and robust across a range of connection probabilities.

Abstract

We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a nonpositive power of the arrival time of $v$, continuing until the graph has $n$ vertices. This class includes uniformly grown random graphs and inhomogeneous random graphs of preferential-attachment type. Whenever the critical percolation threshold is positive, we show that the critical window has width of order $(\log n)^{-2}$ and a secondary phase transition at its finite upper boundary. Inside this window the largest component has size of order $\sqrt{n}/\log n$, and the susceptibility remains finite and independent of the position in the window. The proofs couple component explorations to branching random walks killed outside an interval of length $\log n$, allowing sharp control of the barely subcritical and critical regimes.

The critical percolation window in growing random graphs

TL;DR

We study critical percolation for sparse growing random graphs, introducing the γ-growing model with sequential vertex arrivals and probabilistic connections that mimic preferential attachment. The core method couples component explorations to a two-sided killed branching random walk, enabling precise control of barely subcritical and critical regimes. The main findings establish a critical window of width with the largest component of order and finite, universal susceptibility across the window, plus a secondary transition at the upper boundary. Together, these results delineate a universality class for growing graphs, distinct from mean-field and rank-one models, with a local limit described by the trace of a killed BRW and robust across a range of connection probabilities.

Abstract

We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex with probability proportional to a nonpositive power of the arrival time of , continuing until the graph has vertices. This class includes uniformly grown random graphs and inhomogeneous random graphs of preferential-attachment type. Whenever the critical percolation threshold is positive, we show that the critical window has width of order and a secondary phase transition at its finite upper boundary. Inside this window the largest component has size of order , and the susceptibility remains finite and independent of the position in the window. The proofs couple component explorations to branching random walks killed outside an interval of length , allowing sharp control of the barely subcritical and critical regimes.

Paper Structure

This paper contains 19 sections, 26 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Model definition and main results
  3. Main results
  4. Generalisation to related connection probabilities
  5. Universality landscape
  6. Overview of the proofs
  7. Branching random walks and component exploration
  8. Coupling component exploration with a branching random walk
  9. Local limit is the trace of the $\gamma$-BRW
  10. Preliminaries: Many-to-few formulas
  11. Progeny of the branching random walk killed from two sides
  12. Laplace--Fourier transform of the stopped random walk
  13. Branching random walk
  14. Real, fake and colliding particles
  15. Resolvent of the stopped random walk
  16. ...and 4 more sections

Key Result

Theorem 2.2

Fix $\gamma\in[0, 1/2)$. There exists a positive-valued function $\varepsilon\mapsto M_\varepsilon$ such that for any sequence $\beta_n\to\beta_c$ with $\limsup_{n\to\infty}4\beta_c(\beta_{n}-\beta_c)(\log n)^2<\pi^2$, there exists a constant $n_0$ such that for all $n\ge n_0$ and $\varepsilon>0$,

Theorems & Definitions (63)

  • Definition 2.1: $\gamma$-growing random graph
  • Theorem 2.2: Largest component around criticality
  • Remark 2.3
  • Corollary 2.4: Barely subcritical phase, critical window, and transition across $\pi^2$
  • proof
  • Remark 2.5
  • Conjecture 2.6: Barely supercritical phase
  • Remark 2.7
  • Theorem 2.8: Tail of typical component size
  • Remark 2.9
  • ...and 53 more