Jump to ContentJump to Main Navigation
Random Geometric Graphs$
Users without a subscription are not able to see the full content.

Mathew Penrose

Print publication date: 2003

Print ISBN-13: 9780198506263

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506263.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2019. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 24 August 2019

PERCOLATION AND THE LARGEST COMPONENT

PERCOLATION AND THE LARGEST COMPONENT

Chapter:
(p.194) 10 PERCOLATION AND THE LARGEST COMPONENT
Source:
Random Geometric Graphs
Author(s):

Mathew Penrose (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506263.003.0010

This chapter concerns the largest component of G(N(n),r) in the thermodynamic limit when the number of points N(n) is Poisson with parameter n and the underlying density is uniform on the unit d-cube. It is shown that in the subcritical limit (where the limiting mean degree is below the critical point for continuum percolation), the order of the largest component grows logarithmically with n. In the supercritical limit, the order of the largest component (the ‘giant component’) is asymptotically proportional to n, while the second-largest component grows more slowly, in fact, like the logarithm of the number of points raised to the power d/(d-1). Large deviations and normal approximation results for the order of the largest component are also given.

Keywords:   Poisson, thermodynamic limit, logarithmic growth, giant component, large deviations, normal approximation, second largest component

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .