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A note on dispersing particles on a line

Alan Frieze, Wesley Pegden

TL;DR

It is shown that on the infinite line the final set of occupied sites takes up $O(n)$ space, where $n$ is the number of particles involved in the synchronous dispersion process.

Abstract

We consider a synchronous dispersion process introduced in \cite{CRRS} and we show that on the infinite line the final set of occupied sites takes up $O(n)$ space, where $n$ is the number of particles involved.

A note on dispersing particles on a line

TL;DR

It is shown that on the infinite line the final set of occupied sites takes up space, where is the number of particles involved in the synchronous dispersion process.

Abstract

We consider a synchronous dispersion process introduced in \cite{CRRS} and we show that on the infinite line the final set of occupied sites takes up space, where is the number of particles involved.

Paper Structure

This paper contains 3 sections, 2 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['disperse']}
  3. Remaining proofs

Key Result

Theorem 1

Suppose that we begin the dispersion process on $L$ with $n$ particles at the origin. Then there is an absolute constant $c>0$ such that w.h.p. the furthest particle from the origin is at distance at most $cn$ when the process stops.

Theorems & Definitions (9)

  • Theorem 1
  • Remark
  • proof
  • Lemma 4
  • proof
  • proof
  • proof
  • proof : Proof of Observation \ref{['ends']}
  • proof : Proof of Lemma \ref{['walkvisits']}