Description 
1 online resource (xii, 120 pages) : illustrations (some color). 
Series 
Lecture notes in mathematics, 07215363 ; 2243 

Lecture notes in mathematics, École d'été de probabilités de SaintFlour 

Lecture notes in mathematics (SpringerVerlag) ; 2243.


Lecture notes in mathematics (SpringerVerlag). École d'été de probabilités de SaintFlour.

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 HeSchramm 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 selfcontained 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 singlesemester course; only a basic knowledge of graduate level probability theory is assumed. 
Local Note 
SpringerLink Springer Nature Open Access eBooks 
Subject 
Random walks (Mathematics)


Random walks (Mathematics) (OCoLC)fst01089818

Added Author 
Ecole d'été de probabilités de SaintFlour (48th : 2019 : SaintFlour, France)

ISBN 
9783030279684 (electronic book) 

3030279685 (electronic book) 

9783030279677 (print) 
Standard No. 
10.1007/9783030279684 doi 
