A random walk among random graphs

TL;DR

This paper provides comprehensive lecture notes on various topics in random walks, random graphs, and related probabilistic structures, including proofs of the emergence of the giant component in Erdős–Rényi graphs.

Contribution

It offers an in-depth, pedagogical overview of key concepts and proofs in random graph theory and related stochastic processes, suitable for advanced students.

Findings

Proofs of the emergence of the giant component in Erdős–Rényi graphs

Analysis of random walks and recursive trees

Connections between permutations and continuous-time processes

Abstract

Lecture notes of a master course given at Orsay between 2019-2024. Topics covered include Part I: One-dimensional random walks, cycle lemma and Bienaym\'e--Galton--Watson random trees. Part II: Erd\"os--R\'enyi random graphs, three proofs of the emergence of the giant component. Part III: Random recursive tree, random permutations and continuous time embedding techniques. Intended for publication.