Introduction

We study in this paper the asymptotics as k → ∞ of the motion of a system of k particles located at sites labelled by the integers. This section gives an informal description of the particle system and our results, and the original motivation for the study.

The particles will be referred to as balls, and the sites as boxes. The motion may be described as follows. Initially the k balls are distributed amongst boxes in such a way that the set of occupied boxes is connected. (A box may contain many balls, but there is no empty box between two occupied boxes.) At each move, a ball is taken from the left-most occupied box and placed one box to the right of a ball chosen uniformly at random from among the k balls, the successive choices being mutually independent. It is clear that the set of occupied boxes remains connected, and that the collection of balls drifts off to infinity. It is easy to see that for each k the k-ball motion drifts off to infinity at an almost certain average speed sk, defined formally by (2.3) below. Our main result is that sk ~ e/k as k → ∞. To be more precise:

THEOREM 1.1 As k increases to infinity, kskincreases to e.

This result was conjectured by Tovey (private communication), and informal arguments supporting the conjecture have been given by Keller (1980) and Weiner (1980). Our method of proof (Sections 2–4) is to use coupling to compare the k-ball process with a certain, more easily analysed, pure growth process (defined at (3.3)).