Scaling limits of random graphs

TL;DR

This paper reviews the development of a theory describing the large-scale geometric structure of various random graphs, including trees and the critical Erdős–Rényi model, as their size tends to infinity.

Contribution

It provides a comprehensive overview of the recent advances in the theory of scaling limits for random graphs, highlighting key examples and methods.

Findings

Established the framework for scaling limits of random trees and graphs.

Analyzed the critical Erdős–Rényi random graph within this framework.

Connected geometric structures of large random graphs to their probabilistic properties.

Abstract

This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric structure of these random objects in the limit as their size goes to infinity, with distances appropriately rescaled. We start with the simplest setting of random trees, before turning to various examples of random graphs, including the critical Erd\H{o}s--R\'enyi random graph.