A central study in Probability Theory is the behavior of fluctuation phenomena of partial sums of different types of random variable. One of the most useful concepts for this purpose is that of the random walk which has applications in many areas, particularly in statistical physics and statistical chemistry. Originally published in 1991, Intersections of Random Walks focuses on and explores a number of problems dealing primarily with the nonintersection of random walks and the self-avoiding walk. Many of these problems arise in studying statistical physics and other critical phenomena. Topics Read more... Simple Random Walk -- Harmonic Measure -- Intersection Probabilities -- Four Dimensions -- Two and Three Dimensions -- Self-Avoiding Walks -- Loop-Erased Walk
Author(s): Gregory F Lawler
Series: Modern Birkhäuser classics
Publisher: Birkhauser
Year: 1996, 2013
Language: English
Pages: 226
City: [S.l.]
Tags: Математика;Теория вероятностей и математическая статистика;Теория случайных процессов;
Cover......Page 1
Intersections of Random Walks......Page 4
Preface......Page 8
Notation......Page 14
Contents......Page 16
1.1 Introduction......Page 18
1.2 Local Central Limit Theorem......Page 19
1.3 Strong Markov Property......Page 26
1.4 Harmonic Functions, Dirichlet Problem......Page 28
1.5 Green's Function, Transient Case......Page 35
1.6 Recurrent Case......Page 44
1. 7 Difference Estimates and Harnack Inequality......Page 48
2.1 Definition......Page 54
2.2 Capacity, Transient Case......Page 58
2.3 Capacity, Two Dimensions......Page 64
2.4 Example: Line Segment......Page 69
2.5 Upper Bounds for Harmonic Measure......Page 83
2.6 Diffusion Limited Aggregation......Page 89
3.1 Introduction......Page 94
3.2 Preliminaries......Page 95
3.3 Long Range Intersections......Page 100
3.4 Upper Bound in Four Dimensions......Page 104
3.5 Two-Sided Walks......Page 109
3.6 Upper Bound for Two-Sided Walks......Page 112
3.7 One-sided Walks......Page 120
4.1 Introduction......Page 121
4.2 Two-sided Walks......Page 122
4.3 Long-range Intersections......Page 127
4.4 One-sided Walks......Page 134
4.5 Three Walks in Three Dimensions......Page 140
5.1 Intersection Exponent......Page 144
5.2 Intersections of Brownian Motions......Page 146
5.3 Equivalence of Exponents......Page 151
5.4 Variational Formulation......Page 154
5.5 Lower Bound in Two Dimensions......Page 157
5.6 Upper Bound......Page 160
6.1 Introduction......Page 167
6.2 Connective Constant......Page 168
6.3 Critical Exponents......Page 169
6.4 Edwards Model......Page 174
6.5 Kinetically Growing Walks......Page 179
6.6 Monte Carlo Simulations......Page 182
7.1 Introduction......Page 186
7.2 Erasing Loops......Page 187
7.3 Loop-erased Walk......Page 189
7.4 Two Dimensions......Page 191
7.5 Estimates on Amount Erased......Page 194
7.6 Growth Rate in Low Dimensions......Page 202
7.7 High Dimensions......Page 204
Bibliography......Page 214
Index......Page 219
Appendix A Recent Results......Page 223
Additional References......Page 224