Overview
- 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)
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Table of contents (8 chapters)
Keywords
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.
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
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
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 and Stochastic Processes, Discrete Mathematics, Geometry, Mathematical Physics