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École d'Été de Probabilités de Saint-Flour
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Planar Maps, Random Walks and Circle Packing

École d'Été de Probabilités de Saint-Flour XLVIII - 2018

Authors: Nachmias, Asaf

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  • Entirely self-contained and aimed to fully accompany a single-semester graduate course
  • Many classical proofs have been simplified and streamlined
  • Contains numerous useful exercises
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  • ISBN 978-3-030-27968-4
  • This book is an open access book, you can download it for free on link.springer.com
Softcover $59.99
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  • ISBN 978-3-030-27967-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

This open access book focuses on the interplay between random walks on planar maps and Koebe’s circle packing theorem. Further topics covered include electric networks, the He–Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local limits of finite planar maps and the almost sure recurrence of simple random walks on these limits.  One of its main goals is to present a self-contained proof that the uniform infinite planar triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided.

A planar map is a graph that can be drawn in the plane without crossing edges, together with a specification of the cyclic ordering of the edges incident to each vertex. One widely applicable method of drawing planar graphs is given by Koebe’s circle packing theorem (1936). Various geometric properties of these drawings, such as existence of accumulation points and bounds on the radii, encode important probabilistic information, such as the recurrence/transience of simple random walks and connectivity of the uniform spanning forest. This deep connection is especially fruitful to the study of random planar maps.

The book is aimed at researchers and graduate students in mathematics and is suitable for a single-semester course; only a basic knowledge of graduate level probability theory is assumed.


Table of contents (8 chapters)

Table of contents (8 chapters)
  • Introduction

    Pages 1-10

    Nachmias, Asaf

  • Random Walks and Electric Networks

    Pages 11-31

    Nachmias, Asaf

  • The Circle Packing Theorem

    Pages 33-46

    Nachmias, Asaf

  • Parabolic and Hyperbolic Packings

    Pages 47-60

    Nachmias, Asaf

  • Planar Local Graph Limits

    Pages 61-71

    Nachmias, Asaf

Buy this book

eBook  
  • ISBN 978-3-030-27968-4
  • This book is an open access book, you can download it for free on link.springer.com
Softcover $59.99
price for USA
  • ISBN 978-3-030-27967-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.

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Bibliographic Information

Bibliographic Information
Book Title
Planar Maps, Random Walks and Circle Packing
Book Subtitle
École d'Été de Probabilités de Saint-Flour XLVIII - 2018
Authors
Series Title
École d'Été de Probabilités de Saint-Flour
Series Volume
2243
Copyright
2020
Publisher
Springer International Publishing
Copyright Holder
The Editor(s) (if applicable) and The Author(s)
eBook ISBN
978-3-030-27968-4
DOI
10.1007/978-3-030-27968-4
Softcover ISBN
978-3-030-27967-7
Series ISSN
0721-5363
Edition Number
1
Number of Pages
XII, 120
Number of Illustrations
28 b/w illustrations, 8 illustrations in colour
Topics