Scaling Limits of Random Trees and Random Graphs

Abstract

In the last 30 years, random combinatorial structures and their scaling limits have formed a flourishing area of research at the interface between probability and combinatorics. In this mini-course, I aim to show some of the beautiful theory that arises when considering scaling limits of random trees and graphs. Trees are fundamental objects in combinatorics and the enumeration of different classes of trees is a classical subject. In the first section, we will take as our basic object the genealogical tree of a critical Galton–Watson branching process. (As well as having nice probabilistic properties, this class turns out to include various natural types of random combinatorial tree in disguise.) In the same way as Brownian motion is the universal scaling limit for centred random walks of finite step-size variance, it turns out that all critical Galton–Watson trees with finite offspring variance have a universal scaling limit, Aldous’ Brownian continuum random tree. The simplest model of a random network is the Erdős–Renyi random graph: we take n vertices, and include each possible edge independently with probability p. One of the most well-known features of this model is that it undergoes a phase transition. Take \(p=c/n\). Then for \(c 1\), there is a giant component, comprising a positive fraction of the vertices, and a collection of components of size \(O(\log n)\). (These statements hold with probability tending to 1 as \(n \rightarrow \infty \).) In the second section, we will focus on the critical setting, \(c = 1\), where the largest components have size of order \(n^{2/3}\), and are “close” to being trees, in the sense that they have only finitely many more edges than a tree with the same number of vertices would have. We will see how to use a careful comparison with a branching process in order to derive the scaling limit of the critical Erdős–Renyi random graph. In the final section, we consider the setting of a critical random graph generated according to the configuration model with independent and identically distributed degrees. Here, under natural conditions we obtain the same scaling limit as in the Erdős–Renyi case (up to constants).