Here are two links: the first is to the announcement of my book on the website of CUP, and the second is a postscript file containing table of contents, preface, acknowledgements and the bibliography of my book. It has four chapters, subdivided into 28 Sections, most of which have several sub-sections.

Chapter I, "The type problem", starts with Polya's phenomenom of recurrence of simple random walk on the grids of dimensions one and two versus transience in higher dimensions. As one important feature, it aims at giving a quick and accessible tour to Varopoulos' classification of the recurrent groups; subsequently, recurrence is studied in detail for random walks on various types of structures. Chapter II, "The spectral radius", explains the links between exponential decay of transition probabilities, isoperimetric inequalities, amenability, and explains various methods of computation. Chapter III, "The asymptotic behaviour of transition probabilities", displays a wide range of methods from harmonic analysis via Nash inequalities and related functional analytic tools to singularity analysis and complex functions in order to compute explicit asymptotic equivalents of transition probabilities. In my view, one of the highlights is a careful exposition of Lalley's local limit theorem for finite range random walks on free groups. Chapter IV, "An introduction to topological boundary theory", deals with the interplay between positive harmonic functions and compactifications of the state space. The main problems addressed are the Dirichlet problem at infinity, almost sure convergence to the boundary, and the geometric realization of the Martin boundary. Among other, it contains a recent, elegant proof - comunicated to me by M. Babillot - of the Ney-Spitzer theorem for random walks on integer lattices, and an exposition of Ancona's identification of the Martin boundary of Gromov-hyperbolic graphs.

In this book, I have tried to give careful and ordered expositions of results taken from more than 80 research papers, most of which appeared in the last 20 years and previously were not accessible in any monograph. It is interdisciplinary between several Mathematical disciplines: Probability - Graph Theory - Geometric Group Theory - Discrete Geometry - Discrete Potential Theory - Harmonic Analysis and Spectral Theory.

Chapter I. The type problem

1. Basic facts

2. Recurrence and transience of infinite networks

3. Applications to random walks

4. Isoperimetric inequalities

5. Transient subtrees, and the classification of the recurrent quasi-transitive graphs

6. More on recurrence

Chapter II. The spectral radius

7. Superharmonic functions and rho-recurrence

8. The spectral radius, the rate of escape, and generalized lattices

9. Computing the Green function

10. The spectral radius and strong isoperimetric inequalities

11. A lower bound for simple random walks

12. The spectral radius and amenability

Chapter III. The asymptotic behaviour of transition probabilities

13. The local central limit theorem on the grid

14. Growth, isoperimetric inequalities, and the asymptotic type of random walk

15. The asymptotic type of random walks on amenable groups

16. Simple random walks on the Sierpinski graphs

17. Local limit theorems on free products

18. Intermezzo: Cartesian products

19. Free groups and homogeneous trees

Chapter IV. An introduction to topological boundary theory

20. A probabilistic approach to the Dirichlet problem, and a class of compactifications

21. Ends of graphs and the Dirichlet problem

22. Hyperbolic graphs and groups

23. The Dirichlet problem for circle packing graphs

24. The construction of the Martin boundary

25. Generalized lattices, Abelian and nilpotent groups, and graphs with polynomial growth

26. Trees, ends, and free products

27. The Martin boundary of hyperbolic graphs

28. Cartesian products

Here are the link to my homepage http://www.math.tugraz.at/~woess/

and my email address: woess [at] tugraz [dot] at

Last modified on October 23, 2007