دانلود کتاب Planar maps, random walks and circle packing

فهرست مطالب :

Preface......Page 7
Acknowledgments......Page 8
Contents......Page 10
1.1 The Circle Packing Theorem......Page 12
Infinite Planar Graphs......Page 14
Relation to Conformal Mapping......Page 16
1.2 Probabilistic Applications......Page 18
Recurrence of Distributional Limits of Random Planar Maps......Page 19
2.1 Harmonic Functions and Voltages......Page 22
2.2 Flows and Currents......Page 25
2.3 The Effective Resistance of a Network......Page 28
The Commute Time Identity......Page 32
2.4 Energy......Page 33
2.5 Infinite Graphs......Page 36
2.6 Random Paths......Page 40
2.7 Exercises......Page 41
3.1 Planar Graphs, Maps and Embeddings......Page 43
3.2 Proof of the Circle Packing Theorem......Page 46
Proof of Theorem 3.6, Step 1: Finding the Radii Vector r......Page 47
Proof of Theorem 3.6, Step 2: Drawing the Circle Packing Described by r∞......Page 53
Proof of Theorem 3.6, Step 3: Uniqueness......Page 56
4.1 Infinite Planar Maps......Page 57
4.2 The Ring Lemma and Infinite Circle Packings......Page 58
4.3 Statement of the He–Schramm Theorem......Page 60
Proof of Theorem 4.4 (3)......Page 62
Proof of Theorem 4.4 (4)......Page 63
Proof of Theorem 4.4 (2)......Page 64
4.5 Exercises......Page 69
5.1 Local Convergence of Graphs and Maps......Page 71
Examples......Page 72
5.2 The Magic Lemma......Page 74
Proof of Lemma 5.3......Page 75
5.3 Recurrence of Bounded Degree Planar Graph Limits......Page 77
5.4 Exercises......Page 81
6.1 Star-Tree Transform......Page 82
Stationary Random Graphs......Page 85
Markings......Page 89
6.3 Proof of Theorem 6.1......Page 91
7.1 Introduction......Page 97
Kirchhoff\'s Effective Resistance Formula......Page 99
Spatial Markov Property of the UST......Page 100
7.3 Limits over Exhaustions: The Free and Wired USF......Page 101
7.4 Planar Duality......Page 103
Last Note on Infinite Networks......Page 105
Method of Random Sets......Page 106
Proof of Theorem 7.1......Page 107
7.6 Exercises......Page 111
8 Related Topics......Page 112
References......Page 119
Index......Page 123