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A phase transition in Erdős-Barak random graphs

Gilles Blanchard, Nicolas Curien, Klara Krause, Alexander Reisach

TL;DR

The paper investigates a phase transition for the existence of monotone paths 1 ↗ n in Barak–Erdős random graphs on numbered vertices. It introduces an exploration process that tracks the growth of the reachable set and shows that, after centering and scaling, this process converges to a Gumbel distribution with a deterministic drift, enabling explicit asymptotics for the event 1 ↗ n. The main result identifies a critical window of width Θ(1/n) around p_n = (log n − log log n)/n and provides exact limiting probabilities in terms of a standard Gumbel variable. These findings refine the classical threshold at log n / n and connect the phase transition to Gumbel-type extreme-value limits in monotone-path growth.

Abstract

We study monotone paths in Erdős-Rényi random graphs on numbered vertices. Benjamini & Tzalik established a phase transition at $p = \frac{\log n}{n}$ for this model. We refine the critical value to $p = \frac{\log n - \log \log n }{n}$ and identify the critical window of order $Θ(1/n)$.

A phase transition in Erdős-Barak random graphs

TL;DR

The paper investigates a phase transition for the existence of monotone paths 1 ↗ n in Barak–Erdős random graphs on numbered vertices. It introduces an exploration process that tracks the growth of the reachable set and shows that, after centering and scaling, this process converges to a Gumbel distribution with a deterministic drift, enabling explicit asymptotics for the event 1 ↗ n. The main result identifies a critical window of width Θ(1/n) around p_n = (log n − log log n)/n and provides exact limiting probabilities in terms of a standard Gumbel variable. These findings refine the classical threshold at log n / n and connect the phase transition to Gumbel-type extreme-value limits in monotone-path growth.

Abstract

We study monotone paths in Erdős-Rényi random graphs on numbered vertices. Benjamini & Tzalik established a phase transition at for this model. We refine the critical value to and identify the critical window of order .

Paper Structure

This paper contains 4 sections, 3 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof
  3. Exploration process
  4. Critical window

Key Result

Theorem 1.1

The critical window of the event $\left\{ 1 \nearrow n \textnormal{ in } G_{p_n} \right\}$ is of order $\Theta (1/n)$ around $\frac{\log n - \log \log n}{n}$. More precisely, if $x \in \mathbb{R}$

Figures (1)

  • Figure 1: A segment from a sampled graph $G_p$ satisfying $1 \nearrow 4$ and $1 \nearrow 8$.

Theorems & Definitions (8)

  • Theorem 1.1: Critical window
  • Remark 1.2
  • Proposition 2.1: Convergence of the monotone exploration process
  • proof : Proof of Proposition \ref{['Prop.convergence.to.Gumbel']}
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['Lemma.rest.term']}
  • proof : Proof of Theorem \ref{['Thm.critical.window']}
  • Remark 2.3: Number of paths