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Intersections of random walks. (English) Zbl 0925.60078

Probability and Its Applications. Boston: Birkhäuser. (ISBN 0-8176-3892-X/pbk). 230 p. (1996).
Contents: 1. Simple Random Walk: 1.1 Introduction. 1.2 Local Central Limit Theorem. 1.3 Strong Markov Property. 1.4 Harmonic Functions, Dirichlet Problem. 1.5 Green’s Function, Transient Case. 1.6 Recurrent Case. 1.7 Difference Estimates and Harnack Inequality
2. Harmonic Measure: 2.1 Definition. 2.2 Capacity, Transient Case. 2.3 Capacity, Two Dimensions. 2.4 Example: Line Segment. 2.5 Upper Bound in Four Dimensions. 2.6 Diffusion Limited Aggregation
3. Intersection Probabilities: 3.1 Introduction. 3.2 Preliminaries. 3.3 Long Range Intersections. 3.4 Upper Bound in Four Dimensions. 3.5 Two-Sided Walks. 3.6 Upper Bound for Two-Sided Walks. 3.7 One-Sided Walks
4. Four Dimensions: 4.1 Introduction. 4.2 Two-Sided Walks. 4.3 Long-Range Intersections. 4.4 One-Sided Walks. 4.5 Three Walks in Three Dimensions
5. Two and Three Dimensions: 5.1 Intersection Exponent. 5.2 Intersections of Brownian Motions. 5.3 Equivalence of Exponents. 5.4 Variational Formulas. 5.5 Lower Bound in Two Dimensions. 5.6 Upper Bound
6. Self-Avoiding Walks: 6.1 Introduction. 6.2 Connective Constant. 6.3 Critical Exponents. 6.4 Edwards Model. 6.5 Kinetically Growing Walks. 6.6 Monte Carlo Simulations
7. Loop-Erased Walk: 7.1 Introduction. 7.2 Erasing Loops. 7.3 Loop-Erased Walk. 7.4 Two Dimensions. 7.5 Estimates on Amount Erased. 7.6 Growth Rate in Low Dimensions. 7.7 High Dimensions.

MSC:

60G50 Sums of independent random variables; random walks
60-02 Research exposition (monographs, survey articles) pertaining to probability theory

Citations:

Zbl 1228.60004
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