Random walks and random forests

Abstract

This chapter is inspired by the following quotation from Harris's 1952 paper [193, §6]: Walks and trees. Random walks and branching processes are both objects of considerable interest in probability theory. We may consider a random walk as a probability measure on sequences of steps-that is, on "walks". A branching process is a probability measure on "trees". The purpose of the present section is to show that walks and trees are abstractly identical objects and to give probabilistic consequences of this correspondence. The identity referred to is non-probabilistic and is quite distinct from the fact that a branching process, as a Markov process, may be considered in a certain sense to be a random walk, and also distinct from the fact that each step of the random walk, having two possible directions, represents a twofold branching. This Harris correspondence between walks and trees has been developed and applied in various ways, to enrich the theories of both random walks and branching processes. The chapter is organized as follows: 6.1. Cyclic shifts and Lagrange inversion This section presents a well-known probabilistic interpretation of the Lagrange inversion formula in terms of hitting times of random walks. 6.2. Galton-Watson forests The Lagrange inversion formula appears also in the theory of Galton-Watson branching processes. This is explained by Harris's correspondence between random walk paths on the one hand, and trees or forests on the other. 6.3. Brownian asymptotics for conditioned, Galton-Watson trees The Harris correspondence leads to results due to Aldous and Le Gall, according to which the height profiles of suitably conditioned Galton-Watson trees and forests converge weakly to Brownian excursion or reflecting Brownian bridge, as the number n of vertices in the tree or forest tends to ∞. 6.4. Critical random graphs and the multiplicative coalescent Aldous developed these ideas to obtain the asymptotic behaviour of component sizes of the Erd˝ os-Rényi random graph process G(n, p) in the critical regime p ≈ 1/n as n → ∞. The multiplicative coalescent process governs mergers of connected components in the random graph process as the parameter p increases.