| Description |
1 online resource (xii, 120 pages) : illustrations (some color). |
| Series |
Lecture notes in mathematics, 0721-5363 ; 2243 |
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Lecture notes in mathematics, École d'été de probabilités de Saint-Flour |
|
Lecture notes in mathematics (Springer-Verlag) ; 2243.
|
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Lecture notes in mathematics (Springer-Verlag). École d'été de probabilités de Saint-Flour.
|
| Access |
Open access. GW5XE |
| Bibliography |
Includes bibliographical references and index. |
| Note |
Online resource; title from PDF title page (SpringerLink, viewed October 9, 2019). |
| Contents |
Introduction. -- Random Walks and Electric Networks. -- The Circle Packing Theorem. -- Parabolic and Hyperbolic Packings. -- Planar Local Graph Limits. -- Recurrence of Random Planar Maps. -- Uniform Spanning Trees of Planar Graphs. -- Related Topics. |
| Summary |
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. |
| Local Note |
SpringerLink Springer Nature Open Access eBooks |
| Subject |
Random walks (Mathematics)
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Random walks (Mathematics) (OCoLC)fst01089818
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| Added Author |
Ecole d'été de probabilités de Saint-Flour (48th : 2019 : Saint-Flour, France)
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| ISBN |
9783030279684 (electronic book) |
|
3030279685 (electronic book) |
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9783030279677 (print) |
| Standard No. |
10.1007/978-3-030-27968-4 doi |
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