A random walk on the Rado graph

TL;DR

This paper investigates the mixing time of natural random walks on the Rado graph, revealing that the number of steps needed for convergence grows extremely slowly with the starting point, using Hardy's inequality for trees.

Contribution

It provides the first analysis of random walk mixing times on the Rado graph, establishing bounds involving iterated logarithms and connecting to Hardy's inequality.

Findings

Order log* i steps suffice for convergence from starting point i

For infinitely many i, these steps are necessary

The proof employs Hardy's inequality for trees

Abstract

The Rado graph, also known as the random graph $G(\infty, p)$, is a classical limit object for finite graphs. We study natural ball walks as a way of understanding the geometry of this graph. For the walk started at $i$, we show that order $\log_2^*i$ steps are sufficient, and for infinitely many $i$, necessary for convergence to stationarity. The proof involves an application of Hardy's inequality for trees.