List of publications by Wolfgang Woess

(with some comments)

Papers

1. "Aperiodische Wahrscheinlichkeitsmaße auf topologischen Gruppen", Monatshefte Math. 90 (1980) 339-345.


2. "A local limit theorem for random walks on certain discrete groups", in "Probability Measures on Groups", Oberwolfach 1981, editor: H. Heyer. Springer Lect. Notes in Math. 928 (1982) 467-477.
   
   The first two papers were extracted from my Ph.D. Thesis, entitled "Irrfahrten auf topologischen Gruppen".


3. "Périodicité de mesures de probabilité sur les groupes topologiques", Inst. Elie Cartan 7 (1983) 170-180.
   
   My very first conference talk ever was about papers 1 and 3, in French (I'm not very fluent in French) at a Random Walk conference near Nancy. Two well-known French probabilists sat in the last row chatting all the time, while every now and then interrupting me with a question. After that, I preferred not to give talks in French for many years.
   Paper 3 is on periodicity of irreducible random walks on locally compact groups, including the phenomenum that contrary to discrete groups, no period may exist. When I started to study random walks, I considered it as a basic exercise to understand how periodicity of a random walk on a discrete group is described algebraically. But I learned from a friend that nowadays Ph.D. students in the US are rediscovering this as their own theorems.


4. "Puissances de convolution sur les groupes libres ayant un nombre quelconque de générateurs", Inst. Elie Cartan 7 (1983) 181-190.


5. "Cogrowth of groups and simple random walks", Archiv d. Math. 41 (1983) 363-370.
   This little note has been cited quite often. Among other, it contains a lemma with a much simplified proof of the cogrowth formula of Grigorchuk and Cohen, but I was to shy at the time to state this explicitly, which is why the lemma remained basically unobserved. Later on, variants of that simpler proof were published independently by various people, comprising Szwarc, Northshield, and Bartholdi. Nowadays, for the formula of that lemma, credit is usually given to the (much later) work by Bartholdi, even by authors citing the present old note. (Of course, Laurent Bartholdi should not be blamed for this at all.)


6. "A random walk on free products of finite groups", in "Probability Measures on Groups", Oberwolfach 1983, editor: H. Heyer. Springer Lect. Notes in Math. 1064 (1984) 467-470.


7. "Harmonic functions on free groups", in "Probabilités sur les Structures Géometriques", editor: G. Letac. Proceedings, Toulouse (1984) 141-153.


8. "Chaotic random walks on certain abelian groups", Colloquia Math. Soc. Janos Bolyai 36 (1985) 1147-1167.


9. "Random walks and periodic continued fractions", Advances in Applied Probability 17 (1985) 67-84.


10. with M. A. PICARDELLO: "Random walks on amalgams", Monatshefte Math. 100 (1985) 21-33.


11. with P. GERL: "Simple random walks on trees", European J. Combinatorics 7 (1986) 321-331.


12. with P. GERL: "Local limits and harmonic functions for nonisotropic random walks on free groups", Probability Theory Rel. Fields 71 (1986) 341-355. HIER WEITER


13. "A short computation of the norms of free convolution operators", Proceedings Amer. Math. Soc. 96 (1986) 167-170.
    
    A proof of the Akemann-Ostrand formula via random walk generating functions. Later on, another short proof of the same length was published, allegedly because one of its two authors refused to learn about the generating function technique.


14. "Transience and volumes of trees", Archiv. d. Math. 46 (1986) 184-192.


15. "Nearest neighbour random walks on free products of discrete groups", Bollettino Un. Mat. Ital. 5-B (1986) 961-982.
    
    In this paper, one of my best I believe, the main result is a formula that expresses the generating function of the return probabilites (i.e., the diagonal element of the resolvent) for a random walk on a free product of groups in terms of the resolvents on the free factors. This includes a formula for the radius of convergence (inverse of the spectral radius) and the analytic behaviour near that point, which leads to quite general local limit theorems. At the same time, while I was in Rome, Cartwright and Soardi worked on the same topic in Milan, but none of us was aware of that. They came up with a very similar result (shorter proof, but less general local limit theorem; in reality, I believe that Soardi was the first of us to gain an understanding of the general context), and we became aware of it when having finished. I needed my paper as my "Habilitationsschrift", so I hurried to publish it on my own.
    The submission dates of the two papers are the same, and preceded by several months the submission date of Voiculescu's famous paper on addition of non-comuting random variables, which contains the same formula from a different (more general and more impressive) viewpoint. [Hint: the distributions of the random variables in Voiculescu's setting correspond to the spectral measures of the convolution operators in our framework.]
    As a matter of fact, I sent my reprint twice to Voiculescu and gave a talk on this in his presence, but there was no reaction at all.
    There was a fourth person who found this method independently in the same year, a US Ph.D. Student (John McLaughlin), who used it for computing spectra. This was later on used by Quenell and Gutkin, who in turn did not know much about the other related work.


16. "A description of the Martin boundary for nearest neighbour random walks on free products", in "Probability Measures on Groups", Oberwolfach 1985, editor: H. Heyer. Springer Lect. Notes in Math. 1219 (1986) 203-215.


17. "Harmonic functions on infinite graphs", Rendiconti Sem. Mat. Fis. Milano. 56 (1986) 51-63.


18. with M. A. PICARDELLO: "Martin boundaries of random walks: ends of trees and groups", Transactions Amer. Math. Soc. 302 (1987) 185-205.


19. "Context-free languages and random walks on groups", Discrete Math. 67 (1987) 81-87.


20. "Random walks on infinite graphs", in: "Stochastics in Combinatorial Optimization", CISM Lecture Notes, Udine. editor: G. Andreatta, F. Mason, P. Serafini. World Scientific, Singapore (1987) 255-263.


21. with M. A. PICARDELLO: "Finite truncations of random walks on trees" (appendix to: A. Korànyi, M. A. Picardello, M. Taibleson: "Hardy-spaces on non-homogeneous trees"), Symposia Math. 29 (1988) 255-265.


22. with M. A. PICARDELLO: "Harmonic functions and ends of graphs", Proc. Edinburgh Math. Soc. 31 (1988) 457-461.


23. "Graphs and groups with tree-like properties", J. Combinatorial Th., Ser. B, 47 (1989) 361-371.
    
    In this paper, I proved, among other, the following: a Cayley graph of a finitely generated group is bi-Lipschitz-isometric with a tree iff all ends of the graph are thin iff the group is virtually free. From this it is immediate that a group which is quasi-isometric with a free group is virtually free. I wrote this in January 1986, before Gromov's magnificent essay on Hyperbolic Groups appeared, and submitted it to J. Australian Math. Soc., where it was rejected. I quote from the report:
    "from the group-theoretical point of view it is considered that the main theorem is really about graph theory; there is no application to group theory per se,... . The results are thought to be correct and interesting to the people in 'this' area (probably narrow rather than wide audience)."
    Later, it appeared in JCTB, but - delayed by the rejection - well after Gromov's essay.
    Afterwards, I learned that it was an important theorem of Gromov that a group quasi-isometric with a free group is virtually free. Thus, I learned that it may well happen that a paper by Woess contains not much of interest, while the same result, when done by Gromov, becomes important...


24. with M. A. PICARDELLO: "A converse to the mean value property on homogeneous trees", Transactions Amer. Math. Soc. 311 (1989) 209-225.


25. with B. MOHAR: "A survey on spectra of infinite graphs", Bull. London Math. Soc. 21 (1989) 209-234.


26. with C. D. GODSIL, W. IMRICH, N. SEIFTER and M. E. WATKINS: "On bounded automorphisms of infinite graphs", Graphs and Combinatorics 5 (1989) 333-338.


27. "Amenable group actions on infinite graphs", Math. Annalen. 284 (1989) 251-265.


28. "Boundaries of random walks on graphs and groups with infinitely many ends", Israel J. Math. 68 (1989) 271-301.


29. with M. A. PICARDELLO: "Ends of infinite graphs, potential theory, and electrical networks", in "Cycles and Rays: Basic Structures in Finite and Infinite Graphs", NATO ASI Series, Montréal. editor: G. Hahn, G. Sabidussi, R. E. Woodrow. Kluwer, Dordrecht (1990) 181-196.


30. with P. M. SOARDI: "Amenability, unimodularity, and the spectral radius of random walks on infinite graphs", Math. Zeitschrift 205 (1990) 471-486.


31. with P. M. SOARDI: "Uniqueness of currents in infinite resistive networks", Discrete Applied Math. 31 (1991) 37-49.


32. "Topological groups and infinite graphs", Discrete Math. 95 (1991) 373-384.
    
    At the end of each of the last two papers, I raised the following question: "Is there a vertex-transitive graph that is not quasi-isometric with a Cayley graph?" In an attempt to answer this question, Diestel and Leader invented the DL graphs in the early 1990s, but during the following 13 years, Diestel, Leader, myself and others failed at proving that DL graphs provide the conjectured example. An older and prominent colleague from Geneva seemed to be somewhat ironical about this strange graph-theoretical problem. Some younger people seemed to feel less superior to this type of question (or its querist). I believe that there was a missmatch between the (low) prominence of who posed the question and the (high) difficulty of the problem. In 2004-05, Eskin, Fisher and Whyte produced beautiful work which among other answers that question. Their paper is going to appear in Annals of Math., which induces me to believe that my graph-theoretical question had not been so strange after all.


33. with M. A. PICARDELLO and M. TAIBLESON: "Harmonic functions on Cartesian products of trees with finite graphs", J. Functional Analysis 102 (1991) 379-400.


34. with M. A. PICARDELLO: "Examples of stable Martin boundaries of Markov chains", in "Potential Theory", (M. Kishi, editor), Proceedings, Nagoya 1990, de Gruyter, Berlin (1991) 262-270.


35. "Behaviour at infinity and harmonic functions of random walks on graphs", in "Probability Measures on Groups, X" (H. Heyer, editor), Proceedings, Oberwolfach (1990), Plenum Press, New York (1991) 437-458.


36. with D. I. CARTWRIGHT: "Infinite graphs with nonconstant Dirichlet finite harmonic functions", SIAM J. Discrete Math. 5 (1992) 380-385.


37. with M. A. PICARDELLO and M. TAIBLESON: "Harmonic measure of the planar Cantor set from the viewpoint of graph theory", Discrete Mathematics 109 (1992) 193-202.


38. with V. A. KAIMANOVICH: "Behaviour at infinity and Dirichlet problem for random walks on graphs with a strong isoperimetric inequality", Probability Theory Rel. Fields. 91 (1992) 445-466.


39. with M. A. PICARDELLO: "Martin boundaries of Cartesian products of Markov chains", Nagoya Math. J. 128 (1992), 153-169.


40. with W. IMRICH and N. SAUER: "The average size of nonsingular sets in a graph", in: "Finite and Infinite Combinatorics in Sets and Logic", editor: N. Sauer et al. Kluwer, Dordrecht (1993), 199-205.


41. with D. I. CARTWRIGHT and P. M. SOARDI: "Martin and end compactifications of non locally finite graphs", Transactions Amer. Math. Soc. 338 (1993) 670-693.


42. with C. THOMASSEN: "Vertex-transitive graphs and accessibility", J. Combinatorial Th., Ser. B. 58 (1993) 248-268.


43. "Fixed sets and free subgroups of groups acting on metric spaces", Math. Zeitschrift 214 (1993) 425-440.
    
    When I wrote this, I did not know the important 1987 paper by Gehring and Martin on convergence groups (which originally were discrete groups of quasiconformal mappings). Here, I consider arbitrary isometry groups of a metric space with a boundary at infinity and introduce a property that is analogous to the convergence property. One basic difference is that I distinguish between the group actions on the space and its boundary, which leads to more detailed results. In particular, I get results on isometry groups acting on Gromov hyperbolic spaces. I gave a talk on this at UCLA in 1992, making a bad figure when G. Mess pointed out that "everything" was in the paper of Gehring and Martin. The referee of my paper did not share this point of view, it seems. Nowadays, the proof that the isometry group of a Gromov hyperbolic space acts as a convergence group on the boundary is commonly attributed to a 1994 paper by Tukia. This tells me several things: (1) G. Mess would not have made the same nice remark, had the speaker been Tukia; (2) Even "insiders" of convergence groups did not consider that "everything" was contained in the paper of Gehring and Martin - otherwise why should the Tukia paper be published; (3) the result on hyperbolic spaces is contained in my paper (besides others, and without using the term "convergence group"), but of course nobody among the "insiders" would ever think of giving me any credit or even care to take notice of this paper.


44. with S. GIULINI: "The Martin boundary of the Cartesian product of two hyperbolic spaces", J. Reine Angew. Math. 444 (1993) 17-28.


45. "Random walks on infinite graphs and groups - a survey on selected topics", Bull. London Math. Soc. 26 (1994) 1-60.
    
    It is sometimes claimed that writing survey papers is useless in the sense that it does not count in one's carrer. I do not share this way of thinking. It seems that this survey as well as the one on spectra of infinite graphs with B. Mohar was quite welcome in the respective mathematical communities, and my only minor question has been whether the readers of those surveys noticed that I also have written some research papers...


46. with D. I. CARTWRIGHT and V. A. KAIMANOVICH: "Random walks on the affine group of local fields and of homogeneous trees", Ann. Inst Fourier (Grenoble) 44 (1994) 1243-1288.


47. with M. A. PICARDELLO: "The full Martin boundary of the bi-tree", Annals of Probability 22 (1994) 2203-2222.


48. "Topological groups and recurrence of quasi transitive graphs", Rendiconti Sem. Mat. Fis. Milano 64 (1994) 185-213.


49. "The Martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree", Monatshefte Math. 120 (1995) 55-72.


50. with V. A. KAIMANOVICH: "Construction of discrete, non-unimodular hypergroups", in: Probability Measures on Groups and Related Structures, Proceedings, Oberwolfach 1994 (editor: H. Heyer), World Scientific, Singapore (1995) 196-209.


51. "Dirichlet problem at infinity for harmonic functions on graphs", International Conference on Potential Theory 1994, Proceedings (editors: J. Kral et al.), de Gruyter, Berlin (1996) 189-217.


52. with L. SALOFF-COSTE: "Computing norms of group-invariant transition operators", Combinatorics, Probability, and Computing 5 (1996) 161-178.


53. with P. J. GRABNER: "Functional iterations and periodic oscillations for random walk on the Sierpinski graph", Stochastic Proc. Appl. 69 (1997) 127-138.


54. with L. SALOFF-COSTE: "Transition operators, groups, norms, and spectral radii", Pacific J. Math. 180 (1997) 333-367.


55. "Harmonic functions for group-invariant random walks on graphs", Contemporary Math. 206 (1997) 179-181.


56. "A note on tilings and strong isoperimetric inequality", Math. Proc. Cambridge Phil. Soc. 124 (1998) 385-393.


57. with N. SEIFTER: "Approximating graphs with polynomial growth", Glasgow Math. J. 42 (2000) 1-8.


58. "Irrfahrten", in: Zur Kunst des formalen Denkens (Herausgeber: R. E. Burkard, W. Maass, P. Weibel), Passagen Verlag, Wien (2000) 173-192.


59. with S. BROFFERIO: "On transience of card shuffling", Proc. Amer. Math. Soc. 129 (2001) 1513-1519.


60. "Heat diffusion on homogeneous trees (note on a paper by Medolla and Setti)", Boll. Un. Mat. Ital. 4-B (2001) 703-709 and Erratum (due to printer's error) 5-B (2002) 259-260.
    
    I had an interesting experience with this small note. I had been asked to referee the (analytic) paper it refers to, and saw that a better result was available by applying a well-known, simple probabilistic method to a well known result on the discrete-time analogue. This was pointed out in my report. Afterwards, I learnt that the paper had been published without any change in a different journal than the one for which I was the referee. Slightly irritated, I decided to prepare this note and submit it to that same journal, since obviously there was a need to explain the simple probabilistic method to analysts. But in turn, my paper was rejected there before being published in BUMI !
    Moral: the inbreeding system adopted not only by authors, but also by editors when looking for referees. More precisely: only by chance had I received the paper from the first journal for refereeing, thus being able to point out the simpler proof coming from a different mathematical area. But the responsible editor of the second journal had probably oriented himself via the reference list of the paper, whence it went to another referee who also did not know about the simpler method from a different area, and happily accepted. On the other hand, my subsequent note does contain the correct references from this different area, so that it went to a referee who clearly knew about those simple facts, while of course he could not know what was going on "behind the scene", and refused.
    The story does not end here: after that refusal, I wrote to the editor asking him to comunicate the above observations to the referees of both papers, to let them know about the story "behind the scene". He wrote back accusing me that I wanted to get access to the referee's report of the other paper and to the identity of the referee of my own note !!


61. with V. A. KAIMANOVICH: "Boundary and entropy of space homogeneous Markov chains", Annals of Probability 30 (2002) 323-363.


62. with T. SMIRNOVA-NAGNIBEDA: "Random walks on trees with finitely many cone types", J. Theoret. Probab. 15 (2002) 399-438.
    
    After submitting this paper, we learnt that Steve Lalley had written a paper on the same subject at the same time, proving some of the results (the local limit theorems) under more general assumptions.


63. with T. CECCHERINI-SILBERSTEIN: "Growth and ergodicity of context-free languages", Trans. Amer. Math. Soc. 354 (2002) 4597-4625.


64. with T. CECCHERINI-SILBERSTEIN: "Growth sensitivity of context-free languages", Theoretical Computer Science 307 (2003), 103-116.
    
    This is an extended abstract of the preceding paper. While there, we originally had in mind a mathematical audience, we came to the conclusion that the methods and results should also be presented in an environment of Theoretical Computer Science. Therefore, the presentation and the examples are different here.


65. "Generating function techniques for random walks on graphs", in "Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces", P. Auscher, Th. Coulhon and A. Grigor'yan, eds., Contemporary Math. 338 (2003) 391-423.


66. with D. I. CARTWRIGHT: "Isotropic random walks in a building of type A_d^~ ", Math. Zeitschrift 247 (2004) 101-135.


67. "Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions", Combinatorics, Probability & Computing 14 (2005) 415-433.


68. with L. BARTHOLDI: "Spectral computations on lamplighter groups and Diestel-Leader graphs", J. Fourier Analysis Appl. 11 (2005) 175-202.


69. "A note on the norms of transition operators on lamplighter graphs and groups", Int. J. Algebra and Computation 15 (2005) 1261-1272.


70. with S. BROFFERIO: "Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs", Annales Inst. H. Poincaré (Prob. & Stat.) 41 (2005) 1101-1123.


71. with L. SALOFF-COSTE: "Transition operators on co-compact G-spaces", Revista Matematica Iberoamericana 22 (2006) 747-799.
    
    This is a long paper. I made the experience that the difficulty to publish increases exponentially with the length of a paper. While I believe that it is one of my best in the period 2000-2006, it was rejected more than once on the basis of rather scarse reports (such as I myself would never write). It is much easier to publish 5 papers of 10-15 pages containing just one or two ideas each than one longer one, even though it may have more than 5 times as much substance. Also, good parts of longer papers are not being read except for the first few pages.


72. with S. BROFFERIO: "Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs", Potential Analysis 24 (2006) 245-265.


73. with R. ORTNER: "Non-backtracking random walks and cogrowth of graphs", Canadian J. Math. 59 (2007) 828-844.


74. with A. KARLSSON: "The Poisson boundary of lamplighter random walks on trees", Geometriae Dedicata 124 (2007) 95-107.


75. with D. I. CARTWRIGHT: "The spectrum of the averaging operator on a network (metric graph)", Illinois J. Math. 51 (2007) 805-830.


76. with L. BARTHOLDI and M. NEUHAUSER: "Horocyclic products of trees", J. European Math. Soc. 10 (2008) 771-816.
    
    This is also rather long, but things went well this time...


77. with M. PEIGNÉ: "On recurrence of reflected random walk on the half-line", with an appendix on results of Martin BENDA. Preprint, TU Graz (2006).
    
    Here is some background regarding the appendix: the two unpublished papers cited as [3] and [4] by Benda were submitted in 1999 to PTRF and Ann. Poincaré, respectively. Both were conditionally accepted, requiring revision. The preprints circulated after that, and the ideas were taken up by others. However, the revisions were never carried out and [3,4] remained unpublished. By some detective work, I managed to contact Benda who has quit University and is working for an insurance company now, which is why he lost interest in carrying out the revision. His PhD advisor Kellerer died in 2005, leaving also behind related unpublished work on random walk on the affine group, see the three papers Ergodic Behaviour of Affine Recursions I, II, III on Kellerer's homepage.
    Since we are using Benda's material in the present paper, in particular from [3], we considered it correct to include an appendix referring to him, with brief explanations of those results, so that they might finally become available officially. I asked Martin Benda about it; he agreed.
    Added in December, 2007: it turns out that this paper needed a thorough revision, in order to take into accout two references that we had not been aware of. We will leave the present version on the various webpages but have written a completely new, long paper, see below.


78. with F. LEHNER and M. NEUHAUSER: "On the spectrum of lamplighter groups and percolation clusters", Mathematische Annalen 342 (2008) 69-89.


79. with L. SALOFF-COSTE: "Computations of spectral radii on G-spaces", Contemporary Math. 484 (2009) 195-218.


80. with W. HUSS and E. SAVA: "Entropy sensitivity of languages defined by infinite automata, via Markov chains with forbidden transitions", Theoretical Computer Science 411 (2010) 44-46.


81. with A. BENDIKOV, L. SALOFF-COSTE and M. SALVATORI: "The heat semigroup and Brownian motion on strip complexes", Advances in Mathematics 226 (2011) 992-1055.


82. with T. CECCHERINI-SILBERSTEIN: "Context-free pairs of groups. I - Context-free pairs and graphs", preprint (2009), to appear in European J. Combinatorics.
    This was submitted in November, 2009, to the LMS, but in spite of several reminders, by June 2011 we had no news. Usually I start asking 9 months after submission. In fact, I think it is irresponsible to keep papers so long. I myself have refereed many, many papers for many, many different journals, but never kept a paper for more than 6 months (with one unfortunate exception, where the paper arrived on the very day of my departure to holidays, and on returning, I had forgotten about it). I will not submit to the London Math. Soc. again, nor accept any refereeing for them. Indeed, my experience seems to correspond to a widespread "reputation" of LMS.


83. "Context-free pairs of groups. II - Cuts, tree sets, and random walks", Discrete Mathematics 312 (2012) 157-173.
    This was submitted shortly later to Discrete Mathematics for a proceedings volume honouring Gert Sabidussi's 80th birthday in 2009.


84. with M. PEIGNÉ: "Stochastic dynamical systems with weak contractivity properties, I. Strong and local contractivity. With a chapter featuring results of Martin Benda", Colloquium Math. 125 (2011) 31-54.


85. with M. PEIGNÉ: "Stochastic dynamical systems with weak contractivity properties, II. Iteration of Lipschitz mappings", Colloquium Math. 125 (2011) 55-81.


86. with S. BROFFERIO and M. SALVATORI: "Brownian motion and harmonic functions on Sol(p,q)", Internat. Math. Research Notes, in print.

Books

1. "Catene di Markov e Teoria del Potenziale nel Discreto" (Lecture notes on Markov chains and discrete potential theory), Quaderni dell'Unione Matematica 41 (165 pages), 1996.


2. with M. A. PICARDELLO (editors): "Random Walks and Discrete Potential Theory", Proceedings, Cortona 1997. Symposia Mathematica 39, Cambridge Univ. Press (ix + 361 pages), 1999.


3. "Random Walks on Infinite Graphs and Groups", Cambridge Tracts in Mathematics 138, Cambridge Univ. Press, (xi + 334 pages) 2000; paperback re-edition 2008.


4. with P. M. GRABNER (editors): "Fractals in Graz 2001 - Analysis, Dynamics, Geometry, Stochastics", Proceedings, Graz 2001. Birkhäuser (vii + 283 pages), 2003.


5. with V. A. KAIMANOVICH and K. SCHMIDT (editors): "Random Walks and Geometry", Proceedings of a workshop at the Erwin Schrödinger Insitute in Vienna 2001. De Gruyter, Berlin (x + 532 pages), 2004.


6. "Denumerable Markov Chains - Generating Functions, Boundary Theory, Random Walks on Trees", EMS Textbooks in Mathematics, European Mathematical Society Publihing House (xviii + 351 pages), 2009. (Based on the translation of "Catene di Markov...", but completely rewritten and considerably extended).
   
   As I have experienced with "Random Walks on Infinite Graphs and Groups", the two sets of persons who cite a book, resp. who read parts of a book, are such that neither is a subset of the other. In particular, the presence of some new or improved results - when presented in a book - easily escapes the attention of almost all potential readers. There are such results in both of these books, for example the one on the spectral radius of subdivions of a graph and some stuff on random walks on trees in "Denumerable...". But apparently, hardly anyone has read these parts so far.


7. with D. LENZ and F. SOBIECZKY (editors): "Random Walks, Boundaries and Spectra", Proceedings (Graz - St. Kathrein, 2009), Progess in Probability, vol 64. Birkhäuser, Basel (324 + xxvi pages), 2011.