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  • Open Access
  • © 2020

Planar Maps, Random Walks and Circle Packing

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

Authors:

  • 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

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2243)

Part of the book sub series: École d'Été de Probabilités de Saint-Flour (LNMECOLE)

Buying options

Softcover Book USD 29.99 USD 59.99
50% discount Price excludes VAT (USA)
  • ISBN: 978-3-030-27967-7
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  • Free shipping worldwide
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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xii
  2. Introduction

    • Asaf Nachmias
    Pages 1-10Open Access
  3. Random Walks and Electric Networks

    • Asaf Nachmias
    Pages 11-31Open Access
  4. The Circle Packing Theorem

    • Asaf Nachmias
    Pages 33-46Open Access
  5. Parabolic and Hyperbolic Packings

    • Asaf Nachmias
    Pages 47-60Open Access
  6. Planar Local Graph Limits

    • Asaf Nachmias
    Pages 61-71Open Access
  7. Recurrence of Random Planar Maps

    • Asaf Nachmias
    Pages 73-87Open Access
  8. Uniform Spanning Trees of Planar Graphs

    • Asaf Nachmias
    Pages 89-103Open Access
  9. Related Topics

    • Asaf Nachmias
    Pages 105-111Open Access
  10. Back Matter

    Pages 113-120

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.


Keywords

  • Circle Packing
  • Electric Networks
  • Planar Maps
  • Random Walk
  • Uniform Spanning Trees
  • Open Access

Reviews

“The most remarkable aspect of the Lecture Notes is the reader-friendly structure and the style in which it has been written. There are masses of examples either worked out in the text or left for the reader. A number of facts are equipped with graphical illustrations. The importance of this Lecture Notes by the author both from the practical and from the theoretical standpoint is unquestionable.” (Viktor Ohanyan, zbMATH 1471.60007, 2021)

“The whole material is very nicely presented and the book may serve as the support for a graduate course in probability.” (Nicolas Curien, Mathematical Reviews, November, 2020)

Authors and Affiliations

  • Department of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Asaf Nachmias

Bibliographic Information

  • Book Title: Planar Maps, Random Walks and Circle Packing

  • Book Subtitle: École d'Été de Probabilités de Saint-Flour XLVIII - 2018

  • Authors: Asaf Nachmias

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-27968-4

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s) 2020

  • License: CC BY

  • Softcover ISBN: 978-3-030-27967-7Published: 05 October 2019

  • eBook ISBN: 978-3-030-27968-4Published: 04 October 2019

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XII, 120

  • Number of Illustrations: 28 b/w illustrations, 8 illustrations in colour

  • Topics: Probability Theory, Discrete Mathematics, Geometry, Mathematical Physics

Buying options

Softcover Book USD 29.99 USD 59.99
50% discount Price excludes VAT (USA)
  • ISBN: 978-3-030-27967-7
  • Dispatched in 3 to 5 business days
  • Exclusive offer for individuals only
  • Free shipping worldwide
    See shipping information.
  • Tax calculation will be finalised during checkout