Planar Maps, Random Walks and Circle Packing

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

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Planar Maps, Random Walks and Circle Packing
Asaf Nachmias
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December 29, 2021 | History

Planar Maps, Random Walks and Circle Packing

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

  • 0 Ratings
  • 0 Want to read
  • 0 Currently reading
  • 0 Have read

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.

Publish Date
Language
English

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Edition Availability
Cover of: Planar Maps, Random Walks and Circle Packing
Cover of: Planar Maps, Random Walks and Circle Packing
Planar Maps, Random Walks and Circle Packing
Oct 08, 2020, Saint Philip Street Press
paperback
Cover of: Planar Maps, Random Walks and Circle Packing

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Book Details


Classifications

Library of Congress
QA1-939

The Physical Object

Pagination
xii, 120
Weight
0.454

ID Numbers

Open Library
OL36255564M
ISBN 13
9783030279677

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December 29, 2021 Created by ImportBot Imported from Better World Books record.