Skip to main content Accessibility help
Probability on Trees and Networks
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 75
  • Export citation
  • Recommend to librarian
  • Buy the print book

Book description

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.


‘This long-awaited work focuses on one of the most interesting and important parts of probability theory. Half a century ago, most work on models such as random walks, Ising, percolation and interacting particle systems concentrated on processes defined on the d-dimensional Euclidean lattice. In the intervening years, interest has broadened dramatically to include processes on more general graphs, with trees being a particularly important case. This led to new problems and richer behavior, and as a result, to the development of new techniques. The authors are two of the major developers of this area; their expertise is evident throughout.’

Thomas M. Liggett - University of California, Los Angeles

‘Masterly, beautiful, encyclopaedic, and yet browsable - this great achievement is obligatory reading for anyone working near the conjunction of probability and network theory.’

Geoffrey Grimmett - University of Cambridge

‘For the last ten years, I have not let a doctoral student graduate without reading this [work]. Sadly, the earliest of those students are missing a considerable amount of material that the bound and published edition contains. Not only are the classical topics of random walks, electrical theory, and uniform spanning trees covered in more coherent fashion than in any other source, but this book is also the best place to learn about a number of topics for which the other choices for textual material are limited. These include mass transport, random walk boundaries, and dimension and capacity in the context of Markov processes.’

Robin Pemantle - University of Pennsylvania

‘Lyons and Peres have done an amazing job of motivating their material and of explaining it in a conversational and accessible fashion. Even though the book emphasizes probability on infinite graphs, it is one of my favorite references for probability on finite graphs. If you want to understand random walks, isoperimetry, random trees, or percolation, this is where you should start.’

Daniel Spielman - Yale University, Connecticut

‘This long-awaited book offers a splendid account of several major areas of discrete probability. Both authors have made outstanding contributions to the subject, and the exceptional quality of the book is largely due to their high level of mastery of the field. Although the only prerequisites are basic probability theory and elementary Markov chains, the book succeeds in providing an elegant presentation of the most beautiful and deepest results in the various areas of probability on graphs. The powerful techniques that made these results available, such as the use of isoperimetric inequalities or the mass-transport principle, are also presented in a detailed and self-contained manner. This book will be indispensable to any researcher working in probability on graphs and related topics, and it will also be a must for anybody interested in the recent developments of probability theory.’

Jean-François Le Gall - Université Paris-Sud

'This is a very timely book about a circle of actively developing subjects in discrete probability. No wonder that it became very popular two decades before publication, while still in development. Not only a comprehensive reference source, but also a good textbook to learn the subject, it will be useful for specialists and newcomers alike.'

Stanislav Smirnov - Université of Genève

'A glorious labor of love, compiled over more than two decades of work, that brilliantly surveys the deep and expansive relationships between random trees and other areas of mathematics. Rarely does one encounter a text so exquisitely well written or enjoyable to read. One cannot take more than a few steps in modern probability without encountering one of the topics surveyed here. A truly essential resource.'

Scott Sheffield - Massachusetts Institute of Technology

'There is much to be learned from studying this book. Many of the ideas and tools are useful in a wide variety of different contexts … Geoff Grimmett’s quote on the cover calls the book ‘Masterly, beautiful, encyclopedic and yet browsable.’ I totally agree. Even though it is freely available on the web, you should buy a copy of the book.'

Richard Durrett Source: Mathematical Association of America Reviews (

'This is a monumental book covering a lot of interesting problems in discrete probability, written by two experts in the field … The authors have done a great job of providing full proofs of all main results, hence creating a self-contained reference in this area.'

Abbas Mehrabian Source: Zentralblatt MATH

'This long-awaited book, a project that started in 1993, is bound to be the main reference in the fascinating field of probability on trees and weighted graphs. The authors are the leading experts behind the tremendous developments experienced in the subject in recent decades, where the underlying networks evolved from classical lattices to general graphs … This pedagogically written book is a marvelous support for several courses on topics from combinatorics, Markov chains, geometric group theory, etc., as well as on their inspiring relationships. The wealth of exercises (with comments provided at the end of the book) will enable students and researchers to check their understanding of this fascinating mathematics.'

Laurent Miclo Source: MathSciNet

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.


Abért, M., Glasner, Y., and Virág, B. (2016) The measurable Kesten theorem. Ann. Probab., 44(3), 1601–1646.
Abrams, A. and Kenyon, R. (2015) Fixed-energy harmonic functions. Preprint, :
Achlioptas, D. and McSherry, F. (2001) Fast computation of low rank matrix approximations. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pages 611–618 (electronic). ACM, New York. Held in Hersonissos, 2001, available electronically at MR: 2120364
Achlioptas, D. and McSherry, F. (2007) Fast computation of low-rank matrix approximations. J. ACM, 54(2), Art. 9. MR: 2295993
Adams, S. (1988) Indecomposability of treed equivalence relations. Israel J. Math., 64(3), 362–380. MR: 995576
Adams, S. (1990) Trees and amenable equivalence relations. Ergodic Theory Dynam. Systems, 10(1), 1–14. MR: 91d:28041
Adams, S. and Lyons, R. (1991) Amenability, Kazhdan's property and percolation for trees, groups and equivalence relations. Israel J. Math., 75(2–3), 341–370. MR: 93j:43001
Adams, S. and Spatzier, R.J. (1990) Kazhdan groups, cocycles and trees. Amer. J. Math., 112(2), 271–287. MR: 91c:22011
Adler, J. and Lev, U. (2003) Bootstrap percolation: Visualizations and applications. Brazilian J. Phys., 33(3), 641–644.
Aïdékon, E. (2008) Transient random walks in random environment on a Galton-Watson tree. Probab. Theory Related Fields, 142(3–4), 525–559. MR: 2438700
Aïdékon, E. (2010) Large deviations for transient random walks in random environment on a Galton-Watson tree. Ann. Inst. Henri Poincaré Probab. Stat., 46(1), 159–189. MR: 2641775
Aïdékon, E. (2011) Uniform measure on a Galton-Watson tree without the X log X condition. Preprint,
Aïdékon, E. (2013) Note on the mononicity of the speed of the biased random walk on a Galton-Watson tree.
Aïdékon, E. (2014) Speed of the biased random walk on a Galton-Watson tree. Probab. Theory Related Fields, 159(3–4), 597–617. MR: 3230003
Aizenman, M. (1985) The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory. Comm. Math. Phys., 97(1–2), 91–110. MR: 782960
Aizenman, M. and Barsky, D.J. (1987) Sharpness of the phase transition in percolation models. Comm. Math. Phys., 108(3), 489–526. MR: 88c:82026
Aizenman, M., Burchard, A., Newman, C.M., and Wilson, D.B. (1999) Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms, 15(3–4), 319–367. MR: 2001c:60151
Aizenman, M., Chayes, J.T., Chayes, L., and Newman, C.M. (1988) Discontinuity of the magnetization in one-dimensional Ising and Potts models. J. Statist. Phys., 50(1–2), 1–40. MR: 89f:82072
Aizenman, M., Kesten, H., and Newman, C.M. (1987) Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys., 111(4), 505–531. MR: 89b:82060
Aizenman, M. and Lebowitz, J.L. (1988) Metastability effects in bootstrap percolation. J. Phys. A, 21(19), 3801–3813. MR: 968311
Aldous, D.J (1987) On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probab. Eng. Inform. Sc., 1, 33–46.
Aldous, D.J (1990) The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math., 3(4), 450–465. MR: 91h:60013
Aldous, D.J (1991) Random walk covering of some special trees. J. Math. Anal. Appl., 157(1), 271–283. MR: 1109456
Aldous, D.J and Fill, J.A. (2002) Reversible Markov Chains and Random Walks on Graphs. Unfinished monograph, recompiled 2014 version available at
Aldous, D.J and Lyons, R. (2007) Processes on unimodular random networks. Electron. J. Probab., 12, paper no. 54, 1454–1508 (electronic). MR: 2354165
Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., and Rackoff, C. (1979) Random walks, universal traversal sequences, and the complexity of maze problems. In 20th Annual Symposium on Foundations of Computer Science, pages 218–223. IEEE, New York. Held in San Juan, Puerto Rico, October 29–31, 1979. MR: 598110
Alexander, K.S (1995a) Percolation and minimal spanning forests in infinite graphs. Ann. Probab., 23(1), 87–104. MR: 96c:60114
Alexander, K.S (1995b) Simultaneous uniqueness of infinite clusters in stationary random labeled graphs. Comm. Math. Phys., 168(1), 39–55. Erratum: Comm. Math. Phys. 172, (1995), 221. MR: 96e:60166a
Alexander, K.S and Molchanov, S.A. (1994) Percolation of level sets for two-dimensional random fields with lattice symmetry. J. Statist. Phys., 77(3–4), 627–643. MR: 95i:82052
Alon, N. (1986) Eigenvalues and expanders. Combinatorica, 6(2), 83–96. MR: 88e:05077
Alon, N. (2003) Problems and results in extremal combinatorics. I. Discrete Math., 273(1–3), 31–53. MR: 2025940
Alon, N., Benjamini, I., Lubetzky, E., and Sodin, S. (2007) Non-backtracking random walks mix faster. Commun. Contemp. Math., 9(4), 585–603. MR: 2348845
Alon, N., Benjamini, I., and Stacey, A. (2004) Percolation on finite graphs and isoperimetric inequalities. Ann. Probab., 32(3A), 1727–1745. MR: 2073175
Alon, N., Hoory, S., and Linial, N. (2002) The Moore bound for irregular graphs. Graphs Combin., 18(1), 53–57. MR: 1892433
Alon, N. and Milman, V.D. (1985) isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B, 38(1), 73–88. MR: 782626
Amir, G. and Virág, B. (2016) Speed exponents of random walks on groups. IMRN, 2016.
Anantharam, V. and Tsoucas, P. (1989) A proof of the Markov chain tree theorem. Statist. Probab. Lett., 8(2), 189–192. MR: 1017890
Ancona, A. (1988) Positive harmonic functions and hyperbolicity. In Král, J., Lukes, J., Netuka, I., and Vesely, J., editors, Potential Theory—Surveys and Problems (Prague, 1987), pages 1–23. Springer, Berlin. MR: 973 878
Ancona, A., Lyons, R., and Peres, Y. (1999) Crossing estimates and convergence of Dirichlet functions along random walk and diffusion paths. Ann. Probab., 27(2), 970–989. MR: 1698991
Angel, O., Barlow, M.T., Gurel-Gurevich, O., and Nachmias, A. (2016) Boundaries of planar graphs, via circle packings. Ann. Probab., 44(3), 1956–1984. MR: 3502598
Angel, O. and Benjamini, I. (2007) A phase transition for the metric distortion of percolation on the hypercube. Combinatorica, 27(6), 645–658. MR: 2384409
Angel, O., Benjamini, I., Berger, N., and Peres, Y. (2006) Transience of percolation clusters on wedges. Electron. J. Probab., 11, paper no. 25, 655–669 (electronic). MR: 2242658
Angel, O., Crawford, N., and Kozma, G. (2014) Localization for linearly edge reinforced random walks. Duke Math. J., 163(5), 889–921. MR: 3189433
Angel, O., Friedman, J., and Hoory, S. (2015) The non-backtracking spectrum of the universal cover of a graph. Trans. Amer. Math. Soc., 367(6), 4287–4318. MR: 3324928
Angel, O., Goodman, J., den Hollander, F., and Slade, G. (2008) Invasion percolation on regular trees. Ann. Probab., 36(2), 420–466. MR: 2393988
Angel, O., Goodman, J., and Merle, M. (2013) Scaling limit of the invasion percolation cluster on a regular tree. Ann. Probab., 41(1), 229–261. MR: 3059198
Angel, O. and Szegedy, B. (2016) Recurrence of weak limits of excluded minor graphs. In preparation.
Arratia, R. and Goldstein, L. (2010) Size bias, sampling, the waiting time paradox, and infinite divisibility: When is the increment indepen-dent? Preprint,
Asadpour, A., Goemans, M.X., M'adry, A., Oveis Gharan, S., and Saberi, A. (2010) An n-approximation algorithm for the asymmetric traveling salesman problem. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 379–389. SIAM, Philadelphia. Held in Austin, TX, January 17–19, 2010. MR: 2809683
Asmussen, S. and Hering, H. (1983) Branching Processes. Birkhäuser, Boston. MR: 85b:60076
Athreya, K.B (1971) A note on a functional equation arising in Galton-Watson branching processes. J. Appl. Probability, 8, 589–598. MR: 45:1271
Athreya, K.B and Ney, P.E. (1972) Branching Processes. Vol. 196 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York. MR: 51:9242
Atiyah, M.F (1976) Elliptic operators, discrete groups and von Neumann algebras. In Colloque “Analyse et Topologie” en l'Honneur de Henri Cartan, pages 43–72. Astérisque, 32–33. Soc. Math. France, Paris. Tenu le 17–20 juin 1974 à Orsay. MR: 0420729
Avez, A. (1972) Entropie des groupes de type fini. C. R. Acad. Sci. Paris Sér. A-B, 275, A1363–A1366. MR: 0324741
Avez, A. (1974) Théorème de Choquet-Deny pour les groupes à croissance non exponentielle. C. R. Acad. Sci. Paris Sér. A, 279, 25–28. MR: 0353405
Avez, A. (1976) Croissance des groupes de type fini et fonctions harmoniques. In Théorie Ergodique, Lecture Notes in Mathematics, Vol. 532, pages 35–49. Springer, Berlin. Actes des Journées Ergodiques, Rennes, 1973/1974, Edité par J.-P. Conze et M. S. Keane. MR: 0482911
Azuma, K. (1967) Weighted sums of certain dependent random variables. Tohoku Math. J. (2), 19, 357–367. MR: 0221571
Babai, L. (1991) Local expansion of vertex-transitive graphs and random generation in finite groups. In STOC '91: Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, pages 164–167. ACM, New York.
Babai, L. (1997) The growth rate of vertex-transitive planar graphs. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 564–573. ACM, New York. Held in New Orleans, LA, January 5–7, 1997. MR: 1447704
Babson, E. and Benjamini, I. (1999) Cut sets and normed cohomology with applications to percolation. Proc. Amer. Math. Soc., 127(2), 589–597. MR: 99g:05119
Bahar, I., Atilgan, A.R., and Erman, B. (1997) Direct evaluation of thermal fluctuations in protein using a single parameter harmonic potential. Folding & Design, 2, 173–181.
Bakirov, N.K, Rizzo, M.L., and Székely, G.J. (2006) A multivariate nonparametric test of independence. J. Multivariate Anal., 97(8), 1742–1756. MR: 2298886
Balázs, M. and Folly, A. (2014) Electric network for non-reversible Markov chains. Preprint,
Ball, K. (1992) Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal., 2(2), 137–172. MR: 1159828
Ballmann, W. and Ledrappier, F. (1994) The Poisson boundary for rank one manifolds and their cocompact lattices. Forum Math., 6(3), 301–313. MR: 1269841
Balogh, J., Bollobás, B., Duminil-Copin, H., and Morris, R. (2012) The sharp threshold for bootstrap percolation in all dimensions. Trans. Amer. Math. Soc., 364(5), 2667–2701. MR: 2888224
Balogh, J., Bollobás, B., and Morris, R. (2009) Bootstrap percolation in three dimensions. Ann. Probab., 37(4), 1329–1380. MR: 2546747
Balogh, J., Peres, Y., and Pete, G. (2006) Bootstrap percolation on infinite trees and non-amenable groups. Combin. Probab. Comput., 15(5), 715–730. MR: 2248323
Barlow, M.T (2016) Loop erased walks and uniform spanning trees. In Discrete Geometric Analysis, vol. 34 of MSJ Memoirs, pages 1–32. Mathematical Society of Japan, Tokyo.
Barlow, M.T and Masson, R. (2010) Exponential tail bounds for loop-erased random walk in two dimensions. Ann. Probab., 38(6), 2379–2417. MR: 2683633
Barlow, M.T and Perkins, E.A. (1989) Symmetric Markov chains in Zd: How fast can they move? Probab. Theory Related Fields, 82(1), 95–108. MR: 997432
Barreira, L. (2008) Dimension and Recurrence in Hyperbolic Dynamics. Vol. 272 of Progress in Mathematics. Birkhäuser, Basel. MR: 2434246
Barsky, D.J, Grimmett, G.R., and Newman, C.M. (1991) Percolation in half-spaces: Equality of critical densities and continuity of the percolation probability. Probab. Theory Related Fields, 90(1), 111–148. MR: 92m:60086
Bartholdi, L. (1999) Counting paths in graphs. Enseign. Math. (2), 45(1–2), 83–131. MR: 1703364
Bartholdi, L. (2003) A Wilson group of non-uniformly exponential growth. C. R. Math. Acad. Sci. Paris, 336(7), 549–554. MR: 1981466
Bartholdi, L., Kaimanovich, V.A., and Nekrashevych, V.V. (2010) On amenability of automata groups. Duke Math. J., 154(3), 575–598. MR: 2730578
Bartholdi, L. and Virág, B. (2005) Amenability via random walks. Duke Math. J., 130(1), 39–56. MR: 2176547
Bass, R.F (1995) Probabilistic Techniques in Analysis. Probability and Its Applications. Springer-Verlag, New York. MR: 96e:60001
Bateman, M. and Katz, N.H. (2008) Kakeya sets in Cantor directions. Math. Res. Lett., 15(1), 73–81. MR: 2367175
Beardon, A.F and Stephenson, K. (1990) The uniformization theorem for circle packings. Indiana Univ. Math. J., 39(4), 1383–1425. MR: 1087197
Bekka, M.EB.and Valette, A. (1997) Group cohomology, harmonic functions and the first L2-Betti number. Potential Anal., 6(4), 313–326. MR: 98e:20056
Ben Arous, G., Fribergh, A., Gantert, N., and Hammond, A. (2012) Biased random walks on Galton-Watson trees with leaves. Ann. Probab., 40(1), 280–338. MR: 2917774
Ben Arous, G., Fribergh, A., and Sidoravicius, V. (2014) Lyons-Pemantle-Peres monotonicity problem for high biases. Comm. Pure Appl. Math., 67(4), 519–530. MR: 3168120
Ben Arous, G., Hu, Y., Olla, S., and Zeitouni, O. (2013) Einstein relation for biased random walk on Galton-Watson trees. Ann. Inst. Henri Poincaré Probab. Stat., 49(3), 698–721. MR: 3112431
Benassi, A. (1996) Arbres et grandes déviations. In Chauvin, B., Cohen, S., and Rouault, A., editors, Trees, vol. 40 of Progr. Probab., pages 135–140. Birkhäuser, Basel. Proceedings of the Workshop held in Versailles, June 14–16, 1995. MR: 1439977
Benedetti, R. and Petronio, C. (1992) Lectures on Hyperbolic Geometry. Universitext. Springer-Verlag, Berlin. MR: 94e:57015
Benjamini, I. (1991) Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theoret. Probab., 4(3), 631–637. MR: 1115166
Benjamini, I. and Curien, N. (2012) Ergodic theory on stationary random graphs. Electron. J. Probab., 17, paper no. 93, 20 pp. MR: 2994841
Benjamini, I., Duminil-Copin, H., Kozma, G., and Yadin, A. (2015a) Disorder, entropy and harmonic functions. Ann. Probab., 43(5), 2332–2373.
Benjamini, I., Duminil-Copin, H., Kozma, G., and Yadin, A. (2015b) Minimal harmonic functions I, upper bounds. In preparation.
Benjamini, I., Gurel-Gurevich, O., and Lyons, R. (2007) Recurrence of random walk traces. Ann. Probab., 35(2), 732–738. MR: 2308594
Benjamini, I., Kesten, H., Peres, Y., and Schramm, O. (2004) Geometry of the uniform spanning forest: Transitions in dimensions 4; 8; 12;. Ann. of Math. (2), 160(2), 465–491. MR: 2123930
Benjamini, I. and Kozma, G. (2005) A resistance bound via an isoperimetric inequality. Combinatorica, 25(6), 645–650. MR: 2199429
Benjamini, I., Kozma, G., and Schapira, B. (2011) A balanced excited random walk. C. R. Math. Acad. Sci. Paris, 349(7–8), 459–462. MR: 2788390
Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (1999a) Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab., 27(3), 1347–1356. MR: 1733 151
Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (1999b) Group-invariant percolation on graphs. Geom. Funct. Anal., 9(1), 29–66. MR: 99m:60149
Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (2001) Uniform spanning forests. Ann. Probab., 29(1), 1–65. MR: 1825 141
Benjamini, I., Lyons, R., and Schramm, O. (1999) Percolation perturbations in potential theory and random walks. In Picardello, M. and Woess, W., editors, Random Walks and Discrete Potential Theory, Sympos. Math., XXXIX, pages 56–84. Cambridge University Press, Cambridge. Proceedings of the conference held in Cortona, June 1997. MR: 1802426
Benjamini, I. and Müller, S. (2012) On the trace of branching random walks. Groups Geom. Dyn., 6(2), 231–247. MR: 2914859
Benjamini, I., Nachmias, A., and Peres, Y. (2011) Is the critical percolation probability local? Probab. Theory Related Fields, 149(1–2), 261–269. MR: 2773031
Benjamini, I., Pemantle, R., and Peres, Y. (1995) Martin capacity for Markov chains. Ann. Probab., 23(3), 1332–1346. MR: 96g:60098
Benjamini, I., Pemantle, R., and Peres, Y. (1996) Random walks in varying dimensions. J. Theoret. Probab., 9(1), 231–244. MR: 97a:60092
Benjamini, I., Pemantle, R., and Peres, Y. (1998) Unpredictable paths and percolation. Ann. Probab., 26(3), 1198–1211. MR: 99g:60183
Benjamini, I. and Peres, Y. (1992) Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist., 28(4), 557–592. MR: 94f:60089
Benjamini, I. and Peres, Y. (1994a) Markov chains indexed by trees. Ann. Probab., 22(1), 219–243. MR: 1258875
Benjamini, I. and Peres, Y. (1994b) Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields, 98(1), 91–112. MR: 94m:60141
Benjamini, I. and Schramm, O. (1996a) Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math., 126(3), 565–587. MR: 97k:31009
Benjamini, I. and Schramm, O. (1996b) Percolation beyond Zd, many questions and a few answers. Electron. Comm. Probab., 1, paper no. 8, 71–82 (electronic). MR: 97j:60179
Benjamini, I. and Schramm, O. (1996c) Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab., 24(3), 1219–1238. MR: 98d:60134
Benjamini, I. and Schramm, O. (1997) Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal., 7(3), 403–419. MR: 99b:05032
Benjamini, I. and Schramm, O. (2001a) Percolation in the hyperbolic plane. J. Amer. Math. Soc., 14(2), 487–507. MR: 1815220
Benjamini, I. and Schramm, O. (2001b) Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6, paper no. 23, 13 pp. (electronic). MR: 1873300
Benjamini, I. and Schramm, O. (2004) Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality. In Milman, V.D. and Schechtman, G., editors, Geometric Aspects of Functional Analysis, vol. 1850 of Lecture Notes in Math., pages 73–76. Springer, Berlin. Papers from the Israel Seminar (GAFA) held 2002–2003. MR: 2087152
Bennies, J. and Kersting, G. (2000) A random walk approach to Galton-Watson trees. J. Theoret. Probab., 13(3), 777–803. MR: 1785529
Berestycki, N., Lubetzky, E., Peres, Y., and Sly, A. (2015) Random walks on the random graph. Preprint,
Berger, N., Gantert, N., and Peres, Y. (2003) The speed of biased random walk on percolation clusters. Probab. Theory Related Fields, 126(2), 221–242. MR: 1990055
van den Berg, J. and Keane, M. (1984) On the continuity of the percolation probability function. In Beals, R., Beck, A., Bellow, A., and Hajian, A., editors, Conference in Modern Analysis and Probability (New Haven, Conn., 1982), pages 61–65. Amer. Math. Soc., Providence, RI. MR: 85g:60100
van den Berg, J. and Kesten, H. (1985) Inequalities with applications to percolation and reliability. J. Appl. Probab., 22(3), 556–569. MR: 87b:60027
van den Berg, J. and Kesten, H. (1991) Stability properties of a flow process in graphs. Random Structures Algorithms, 2(3), 335–341. MR: 92d:90027
Biggins, J.D (1977) Chernoff's theorem in the branching random walk. J. Appl. Probability, 14(3), 630–636. MR: 0464415
Biggs, N.L, Mohar, B., and Shawe-Taylor, J. (1988) The spectral radius of infinite graphs. Bull. London Math. Soc., 20(2), 116–120. MR: 89a:05103
Billingsley, P. (1965) Ergodic Theory and Information. John Wiley, New York. MR: 33:254
Billingsley, P. (1995) Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. John Wiley, New York. MR: 1324786
Björklund, M. (2014) Five remarks about random walks on groups. Preprint,
Blachère, S., Haïssinsky, P., and Mathieu, P. (2008) Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab., 36(3), 1134–1152. MR: 2408585
Blackwell, D. (1955) On transient Markov processes with a countable number of states and stationary transition probabilities. Ann. Math. Statist., 26, 654–658. MR: 17,754d
Boley, D., Ranjan, G., and Zhang, Z.L. (2011) Commute times for a directed graph using an asymmetric Laplacian. Linear Algebra Appl., 435(2), 224–242. MR: 2782776
Bollobás, B. (1998) Modern Graph Theory. Vol. 184 of Graduate Texts in Mathematics. Springer-Verlag, New York. MR: 99h:05001
Borcea, J., Brändén, P., and Liggett, T.M. (2009) Negative dependence and the geometry of polynomials. J. Amer. Math. Soc., 22(2), 521–567. MR: 2476782
Borre, K. and Meissl, P. (1974) Strength analysis of leveling-type networks. An application of random walk theory. Geodaet. Inst. Medd., 50, 80. MR: 0475698
Borwein, J.M and Zucker, I.J. (1992) Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal., 12(4), 519–526. MR: 1186733
Bourgain, J. (1985) On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math., 52(1–2), 46–52. MR: 815600
Bowen, L. (2004) Couplings of uniform spanning forests. Proc. Amer. Math. Soc., 132(7), 2151–2158 (electronic). MR: 2053 989
Brieussel, J. (2009) Amenability and non-uniform growth of some directed automorphism groups of a rooted tree. Math. Z., 263(2), 265–293. MR: 2534118
Brieussel, J. (2013) Behaviors of entropy on finitely generated groups. Ann. Probab., 41(6), 4116–4161. MR: 3161471
Brieussel, J. and Zheng, T. (2015) Speed of random walks, isoperimetry and compression of finitely generated groups. Preprint,
Broadbent, S.R and Hammersley, J.M. (1957) Percolation processes. I. Crystals and mazes. Proc. Cambridge Philos. Soc., 53, 629–641. MR: 0091567
Broder, A. (1989) Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science (Research Triangle Park, North Carolina), pages 442–447. IEEE, New York.
Broder, A.Z and Karlin, A.R. (1989) Bounds on the cover time. J. Theoret. Probab., 2(1), 101–120. MR: 981768
Brofferio, S. and Schapira, B. (2011) Poisson boundary of GLd. Israel J. Math., 185, 125–140. MR: 2837130
Broman, E.I, Camia, F., Joosten, M., and Meester, R. (2013) Dimension (in)equalities and Hölder continuous curves in fractal percolation. J. Theoret. Probab., 26(3), 836–854. MR: 3090553
Brooks, R.L, Smith, C.A.B., Stone, A.H., and Tutte, W.T. (1940) The dissection of rectangles into squares. Duke Math. J., 7, 312–340. MR: 2,153d
Burton, R.M and Keane, M. (1989) Density and uniqueness in percolation. Comm. Math. Phys., 121(3), 501–505. MR: 90g:60090
Burton, R.M and Pemantle, R. (1993) Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer- impedances. Ann. Probab., 21(3), 1329–1371. MR: 94m:60019
Buser, P. (1982) A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4), 15(2), 213–230. MR: 683635
Campanino, M. (1985) Inequalities for critical probabilities in percolation. In Durrett, R., editor, Particle Systems, Random Media and Large Deviations, vol. 41 of Contemp. Math., pages 1–9. Amer. Math. Soc., Providence, RI. Proceedings of the AMS-IMS-SIAM joint summer research conference in the mathematical sciences on mathematics of phase transitions held at Bowdoin College, Brunswick, Maine, June 24–30, 1984. MR: 814699
Campbell, G.A (1911) Cisoidal oscillations. Trans. Amer. Inst. Electrical Engineers, 30, 873–909.
Cannon, J.W, Floyd, W.J., Kenyon, R.W., and Parry, W.R. (1997) Hyperbolic geometry. In Levy, S., editor, Flavors of Geometry, vol. 31 of Mathematical Sciences Research Institute Publications, pages 59–115. Cambridge University Press, Cambridge. MR: 1491098
Cannon, J.W, Floyd, W.J., and Parry, W.R. (1994) Squaring rectangles: The finite Riemann mapping theorem. In Abikoff, W., Birman, J.S., and Kuiken, K., editors, The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions, vol. 169 of Contemp. Math., pages 133–212. Amer. Math. Soc., Providence, RI. Proceedings of the conference held at Polytechnic University, Brooklyn, New York, May 1–3, 1992. MR: 1292901
Caputo, P., Liggett, T.M., and Richthammer, T. (2010) Proof of Aldous' spectral gap conjecture. J. Amer. Math. Soc., 23(3), 831–851. MR: 2629990
Carleson, L. (1967) Selected Problems on Exceptional Sets. Vol. 13 of Van Nostrand Mathematical Studies. D. Van Nostrand, Princeton, NJ. MR: 0225986
Carlitz, L., Wilansky, A., Milnor, J., Struble, R.A., Felsinger, N., Simoes, J.M.S., Power, E.A., Shafer, R.E., and Maas, R.E. (1968) Problems and Solutions: Advanced Problems: 5600–5609. Amer. Math. Monthly, 75(6), 685–687. MR: 1534960
Carmesin, J. (2012) A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy. Potential Anal., 37(3), 229–245. MR: 2969301
Carmesin, J. and Georgakopoulos, A. (2015) Every planar graph with the Liouville property is amenable. Preprint,
Carne, T.K (1985) A transmutation formula for Markov chains. Bull. Sci. Math. (2), 109(4), 399–405. MR: 87m:60142
Cartwright, D.I, Kaimanovich, V.A., and Woess, W. (1994) Random walks on the affine group of local fields and of homogeneous trees. Ann. Inst. Fourier (Grenoble), 44(4), 1243–1288. MR: 1306556
Cartwright, D.I and Soardi, P.M. (1989) Convergence to ends for random walks on the automorphism group of a tree. Proc. Amer. Math. Soc., 107(3), 817–823. MR: 90f:60137
Cayley, A. (1889) A theorem on trees. Quart. J. Math., 23, 376–378.
Teixeira, A.Q. (2012) From Random Walk Trajectories to Random Interlacements. Vol. 23 of Ensaios Matemáticos [Mathe- matical Surveys]. Sociedade Brasileira de Matemática, Rio de Janeiro. MR: 3014964
Chaboud, T. and Kenyon, C. (1996) Planar Cayley graphs with regular dual. Internat. J. Algebra Comput., 6(5), 553–561. MR: 98a:05077
Chalupa, J., Reich, G.R., and Leath, P.L. (1979) Bootstrap percolation on a Bethe lattice. J. Phys. C, 12(1), L31–L35. 0022-3719/12/1/008.
Chandra, A.K, Raghavan, P., Ruzzo, W.L., Smolensky, R., and Tiwari, P. (1996/1997) The electrical resistance of a graph captures its commute and cover times. Comput. Complexity, 6(4), 312–340. MR: 99h:60140
Chauvin, B., Rouault, A., and Wakolbinger, A. (1991) Growing conditioned trees. Stochastic Process. Appl., 39(1), 117–130. MR: 1135089
Chayes, J.T, Chayes, L., and Durrett, R. (1988) Connectivity properties of Mandelbrot's percolation process. Probab. Theory Related Fields, 77(3), 307–324. MR: 931500
Chayes, J.T, Chayes, L., and Newman, C.M. (1985) The stochastic geometry of invasion percolation. Comm. Math. Phys., 101(3), 383–407. MR: 87i:82072
Chayes, L. (1995a) Aspects of the fractal percolation process. In Bandt, C., Graf, S., and Zähle, M., editors, Fractal Geometry and Stochastics, vol. 37 of Progr. Probab., pages 113–143. Birkhäuser, Basel. Papers from the conference held in Finsterbergen, June 12–18, 1994. MR: 1391973
Chayes, L. (1995b) On the absence of directed fractal percolation. J. Phys. A.: Math. Gen., 28, L295–L301.
Chayes, L., Pemantle, R., and Peres, Y. (1997) No directed fractal percolation in zero area. J. Statist. Phys., 88(5–6), 1353–1362. MR: 1478072
Cheeger, J. (1970) A lower bound for the smallest eigenvalue of the Laplacian. In Gunning, R.C., editor, Problems in Analysis, pages 195–199. Princeton University Press, Princeton, NJ. A symposium in honor of Salomon Bochner, Princeton University, Princeton, NJ, 1–3 April 1969. MR: 53:6645
Cheeger, J. and Gromov, M. (1986) L2-cohomology and group cohomology. Topology, 25(2), 189–215. MR: 87i:58161
Chen, D. (1997) Average properties of random walks on Galton-Watson trees. Ann. Inst. H. Poincaré Probab. Statist., 33(3), 359–369. MR: 1457056
Chen, D. and Peres, Y. (2004) Anchored expansion, percolation and speed. Ann. Probab., 32(4), 2978–2995. With an appendix by Gábor Pete. MR: 2094436
Chen, D. and Zhang, F.X. (2007) On the monotonicity of the speed of random walks on a percolation cluster of trees. Acta Math. Sin. (Engl. Ser.), 23(11), 1949–1954. MR: 2359112
Choquet, G. and Deny, J. (1960) Sur l'équation de convolution. C. R. Acad. Sci. Paris, 250, 799–801. MR: 22:9808
Chung, F.RK., Graham, R.L., Frankl, P., and Shearer, J.B. (1986) Some intersection theorems for ordered sets and graphs. J. Combin. Theory Ser. A, 43(1), 23–37. MR: 859293
Chung, F.RK.and Tetali, P. (1998) Isoperimetric inequalities for Cartesian products of graphs. Combin. Probab. Comput., 7(2), 141–148. MR: 2000c:05085
Colbourn, C.J, Provan, J.S., and Vertigan, D. (1995) A new approach to solving three combinatorial enumeration problems on planar graphs. Discrete Appl. Math., 60(1–3), 119–129. MR: 96e:05154
Collevecchio, A. (2006) On the transience of processes defined on Galton-Watson trees. Ann. Probab., 34(3), 870–878. MR: 2243872
Constantine, G.M (2003) Graphs, networks, and linear unbiased estimates. Discrete Appl. Math., 130(3), 381–393. MR: 1999697
Coppersmith, D., Doyle, P., Raghavan, P., and Snir, M. (1993) Random walks on weighted graphs and applications to on-line algorithms. J. Assoc. Comput. Mach., 40(3), 421–453. MR: 1370357
Coppersmith, D., Feige, U., and Shearer, J. (1996) Random walks on regular and irregular graphs. SIAM J. Discrete Math., 9(2), 301–308. MR: 1386885
Coppersmith, D., Tetali, P., and Winkler, P. (1993) Collisions among random walks on a graph. SIAM J. Discrete Math., 6(3), 363–374. MR: 1229691
Coulhon, T. (1996) Ultracontractivity and Nash type inequalities. J. Funct. Anal., 141(2), 510–539. MR: 1418518
Coulhon, T., Grigor'yan, A., and Pittet, C. (2001) A geometric approach to on-diagonal heat kernel lower bounds on groups. Ann. Inst. Fourier (Grenoble), 51(6), 1763–1827. MR: 1871289
Coulhon, T. and Saloff-Coste, L. (1993) Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana, 9(2), 293–314. MR: 94g:58263
Cournot, A.A (1847) De l'origine et des limites de la correspondance entre l'algèbre et la géométrie. Hachette, Paris. Available at
Cox, J.T and Durrett, R. (1983) Oriented percolation in dimensions: Bounds and asymptotic formulas. Math. Proc. Cambridge Philos. Soc., 93(1), 151–162. MR: 84e:60150
Croydon, D. (2008) Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. Henri Poincaré Probab. Stat., 44(6), 987–1019. MR: 2469332
Croydon, D. and Kumagai, T. (2008) Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive. Electron. J. Probab., 13, paper no. 51, 1419–1441. MR: 2438812
Curien, N. and Le Gall, J.F. (2011) Random recursive triangulations of the disk via fragmentation theory. Ann. Probab., 39(6), 2224–2270. MR: 2932668
Curien, N. and Le Gall, J.F. (2016) The harmonic measure of balls in random trees. Ann. Probab. To appear,
Curien, N. and Peres, Y. (2011) Random laminations and multitype branching processes. Electron. Commun. Probab., 16, 435–446. MR: 2831082
Dai, J.J (2005) A once edge-reinforced random walk on a Galton-Watson tree is transient. Statist. Probab. Lett., 73(2), 115–124. MR: 2159246
Davenport, H., Erdʺos, P., and LeVeque, W.J. (1963) On Weyl's criterion for uniform distribution. Michigan Math. J., 10, 311–314. MR: 0153656
Dekking, F.M and Meester, R.W.J. (1990) On the structure of Mandelbrot's percolation process and other random Cantor sets. J. Statist. Phys., 58(5–6), 1109–1126. MR: 1049059
Dellacherie, C. and Meyer, P.A. (1978) Probabilities and Potential. Vol. 29 of North-Holland Mathematics Studies. North-Holland, Amsterdam. MR: 521810
Dembo, A. (2005) Favorite points, cover times and fractals. In Lectures on Probability Theory and Statistics, vol. 1869 of Lecture Notes in Math., pages 1–101. Springer, Berlin. Lectures from the 33rd Probability Summer School held in Saint-Flour, July 6–23, 2003, edited by Jean Picard. MR: 2228383
Dembo, A., Gantert, N., Peres, Y., and Zeitouni, O. (2002) Large deviations for random walks on Galton-Watson trees: Averaging and uncertainty. Probab. Theory Related Fields, 122(2), 241–288. MR: 1894069
Dembo, A., Gantert, N., and Zeitouni, O. (2004) Large deviations for random walk in random environment with holding times. Ann. Probab., 32(1B), 996–1029. MR: 2044672
Dembo, A., Peres, Y., Rosen, J., and Zeitouni, O. (2001) Thick points for planar Brownian motion and the Erdʺos-Taylor conjecture on random walk. Acta Math., 186(2), 239–270. MR: 1846031
Dembo, A. and Zeitouni, O. (1998) Large Deviations Techniques and Applications, 2nd ed. Vol. 38 of Applications of Mathematics. Springer-Verlag, New York. MR: 1619036
Derriennic, Y. (1976) Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires. Ann. Inst. H. Poincaré Sect. B (N.S.), 12(2), 111–129. MR: 0423532
Derriennic, Y. (1980) Quelques applications du théorème ergodique sous-additif. In Journées sur les Marches Aléatoires, vol. 74 of Astérisque, pages 183–201, 4. Soc. Math. France, Paris. Held at Kleebach, March 5–10, 1979. MR: 588163
Deza, M.M and Laurent, M. (1997) Geometry of Cuts and Metrics. Vol. 15 of Algorithms and Combinatorics. Springer-Verlag, Berlin. MR: 1460488
Diaconis, P. and Evans, S.N. (2002) A different construction of Gaussian fields from Markov chains: Dirichlet covariances. Ann. Inst. H. Poincaré Probab. Statist., 38(6), 863–878. MR: 1955341
Diaconis, P. and Fill, J.A. (1990) Strong stationary times via a new form of duality. Ann. Probab., 18(4), 1483–1522. MR: 1071805
Diaconis, P. and Saloff-Coste, L. (1994) Moderate growth and random walk on finite groups. Geom. Funct. Anal., 4(1), 1–36. MR: 1254308
Diestel, R. and Leader, I. (2001) A conjecture concerning a limit of non-Cayley graphs. J. Algebraic Combin., 14(1), 17–25. MR: 2002h:05082
Ding, J., Lee, J.R., and Peres, Y. (2012) Cover times, blanket times, and majorizing measures. Ann. of Math. (2), 175(3), 1409–1471. MR: 2912708
Ding, J., Lee, J.R., and Peres, Y. (2013) Markov type and threshold embeddings. Geom. Funct. Anal., 23(4), 1207–1229. MR: 3077911
Disertori, M., Sabot, C., and Tarrès, P. (2015) Transience of edge-reinforced random walk. Comm. Math. Phys., 339(1), 121–148. MR: 3366053
Dodziuk, J. (1977) de Rham-Hodge theory for L2-cohomology of infinite coverings. Topology, 16(2), 157–165. MR: 0445560
Dodziuk, J. (1979) L2 harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Amer. Math. Soc., 77(3), 395–400. MR: 81e:58004
Dodziuk, J. (1984) Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc., 284(2), 787–794. MR: 85m:58185
Dodziuk, J. and Kendall, W.S. (1986) Combinatorial Laplacians and isoperimetric inequality. In Elworthy, K.D., editor, From Local Times to Global Geometry, Control and Physics, pages 68–74. Longman Sci. Tech., Harlow. Papers from the Warwick symposium on stochastic differential equations and applications, held at the University of Warwick, Coventry, 1984/85. MR: 88h:58118
Doob, J.L (1984) Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, New York. MR: 85k:31001
Doyle, P.G (1988) Electric currents in infinite networks. Unpublished,
Doyle, P.G (2009) The Kemeny constant of a Markov chain. Unpublished,
Doyle, P.G and Snell, J.L. (1984) Random Walks and Electric Networks. Mathematical Association of America, Washington, DC. Also available at MR: 89a:94023
Doyle, P.G and Steiner, J. (2008) Commuting time geometry of ergodic Markov chains. Unpublished,
Drewitz, A., Ráth, B., and Sapozhnikov, A. (2014) An Introduction to Random Interlacements. Springer Briefs in Mathematics. Springer, Cham. MR: 3308116
Dubins, L.E and Freedman, D.A. (1967) Random distribution functions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), pages Vol. II: Contributions to Probability Theory, Part 1, pp. 183–214. University of California Press, Berkeley. MR: 0214109
Duffin, R.J (1962) The extremal length of a network. J. Math. Anal. Appl., 5, 200–215. MR: 0143468
Duminil-Copin, H., Sidoravicius, V., and Tassion, V. (2016) Absence of infinite cluster for critical Bernoulli percolation on slabs. Comm. Pure Appl. Math., 69(7), 1397–1411. MR: 3503025
Duminil-Copin, H. and Smirnov, S. (2012) The connective constant of the honeycomb lattice equals. Ann. of Math. (2), 175(3), 1653–1665. MR: 2912714
Duminil-Copin, H. and Tassion, V. (2015) A new proof of the sharpness of the phase transition for Bernoulli percolation on.d. Enseign. Math. To appear,
Duminil-Copin, H. and Tassion, V. (2016) A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Comm. Math. Phys., 343(2), 725–745. MR: 3477351
Duplantier, B. (1992) Loop-erased self-avoiding walks in 2D. Physica A, 191, 516–522. 0378-4371(92)90575-B.
Duquesne, T. and Le Gall, J.F. (2002) Random trees, Lévy processes and spatial branching processes. Astérisque, 281, vi+147. MR: 1954248
Durrett, R. (1986) Reversible diffusion processes. In Chao, J.A. and Woyczyński, W.A., editors, Probability Theory and Harmonic Analysis, pages 67–89. Dekker, New York. Papers from the conference held in Cleveland, Ohio, May 10–12, 1983. MR: 88b:60175
Durrett, R. (2010) Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. MR: 2722836
Durrett, R., Kesten, H., and Lawler, G. (1991) Making money from fair games. In Durrett, R. and Kesten, H., editors, Random Walks, Brownian Motion, and Interacting Particle Systems, vol. 28 of Progr. Probab., pages 255–267. Birkhäuser, Boston. A Festschrift in honor of Frank Spitzer. MR: 1146451
Dvoretzky, A. (1949) On the strong stability of a sequence of events. Ann. Math. Statistics, 20, 296–299. MR: 0031675
Dvoretzky, A. and Erdös, P. (1951) Some problems on random walk in space. In Proc. Second Berkeley Symposium on Math. Statist. and Probability, 1950, pages 353–367. University of California Press, Berkeley. MR: 0047272
Dvoretzky, A., Erdös, P., and Kakutani, S. (1950) Double points of paths of Brownian motion in n-space. Acta Sci. Math. Szeged, 12(Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B), 75–81. MR: 0034972
Dvoretzky, A., Erdʺos, P., Kakutani, S., and Taylor, S.J. (1957) Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc., 53, 856–862. MR: 0094855
Dwass, M. (1969) The total progeny in a branching process and a related random walk. J. Appl. Probability, 6, 682–686. MR: 0253433
Dymarz, T. (2010) Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups. Duke Math. J., 154(3), 509–526. MR: 2730576
Dynkin, E.B (1969) The boundary theory of Markov processes (discrete case). Uspehi Mat. Nauk, 24(2), 3–42. English translation: Russ. Math. Surv. 24 (1969), no. 2, 1–42; MR: 0245096
Dynkin, E.B (1980) Markov processes and random fields. Bull. Amer. Math. Soc. (N.S.), 3(3), 975–999. MR: 585179
Dynkin, E.B and Maljutov, M.B. (1961) Random walk on groups with a finite number of generators. Dokl. Akad. Nauk SSSR, 137, 1042–1045. English translation: Soviet Math. Dokl. (1961) 2, 399–402. MR: 24:A1751
Eckmann, B. (2000) Introduction to l2-methods in topology: Reduced l2-homology, harmonic chains, l2-Betti numbers. Israel J. Math., 117, 183–219. Notes prepared by Guido Mislin. MR: 1760592
Edgar, G.A (1990) Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. Springer-Verlag, New York. MR: 1065392
Edmonds, J. (1971) Matroids and the greedy algorithm. Math. Programming, 1, 127–136. MR: 0297357
Enflo, P. (1969) On the nonexistence of uniform homeomorphisms between Lp-spaces. Ark. Mat., 8, 103–105 (1969). MR: 0271719
Epstein, I. and Hjorth, G. (2009) Rigidity and equivalence relations with infinitely many ends. Unpublished manuscript available at
Erschler, A. (2010) Poisson-Furstenberg boundaries, large-scale geometry and growth of groups. In Proceedings of the International Congress of Mathematicians. Volume II, pages 681–704. Hindustan Book Agency, New Delhi. MR: 2827814
Erschler, A. (2011) Poisson-Furstenberg boundary of random walks on wreath products and free metabelian groups. Comment. Math. Helv., 86(1), 113–143. MR: 2745278
Erschler, A. and Karlsson, A. (2010) Homomorphisms to. constructed from random walks. Ann. Inst. Fourier (Grenoble), 60(6), 2095–2113. MR: 2791651
Eskin, A., Fisher, D., and Whyte, K. (2012) Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs. Ann. of Math. (2), 176(1), 221–260. MR: 2925383
Ethier, S.N and Lee, J. (2009) Limit theorems for Parrondo's paradox. Electron. J. Probab., 14, paper no. 62, 1827–1862. MR: 2540850
Evans, S.N (1987) Multiple points in the sample paths of a Lévy process. Probab. Theory Related Fields, 76(3), 359–367. MR: 912660
Falconer, K.J (1986) Random fractals. Math. Proc. Cambridge Philos. Soc., 100(3), 559–582. MR: 88e:28005
Falconer, K.J (1987) Cut-set sums and tree processes. Proc. Amer. Math. Soc., 101(2), 337–346. MR: 88m:90052
Falconer, K.J (1990) Fractal Geometry. Mathematical foundations and applications. John Wiley, Chichester. MR: 1102677
Fan, A.H (1989) Décompositions de Mesures et Recouvrements Aléatoires. Ph.D. thesis, Université de Paris-Sud, Département de Mathématique, Orsay. MR: 91e:60009
Fan, A.H (1990) Sur quelques processus de naissance et de mort. C. R. Acad. Sci. Paris Sér. I Math., 310(6), 441–444. MR: 91d:60103
Faraud, G. (2011) A central limit theorem for random walk in a random environment on marked Galton-Watson trees. Electron. J. Probab., 16, paper no. 6, 174–215. MR: 2754802
Faraud, G., Hu, Y., and Shi, Z. (2012) Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Related Fields, 154(3–4), 621–660. MR: 3000557
Feder, T. and Mihail, M. (1992) Balanced matroids. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pages 26–38. Association for Computing Machinery (ACM), New York.
Feller, W. (1971) An Introduction to Probability Theory and Its Applications. Vol. II., 2nd ed. John Wiley, New York. MR: 42:5292
Fisher, M.E (1961) Critical probabilities for cluster size and percolation problems. J. Math. Phys., 2, 620–627. MR: 0126306
Fitzner, R. and van der Hofstad, R. (2015) Nearest-neighbor percolation function is continuous for d > 10. Preprint,
Fitzsimmons, P.J and Salisbury, T.S. (1989) Capacity and energy for multiparameter Markov processes. Ann. Inst. H. Poincaré Probab. Statist., 25(3), 325–350. MR: 1023955
Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics. Cambridge University Press, Cambridge. MR: 2483235
Flanders, H. (1971) Infinite networks. I: Resistive networks. IEEE Trans. Circuit Theory, CT–18, 326–331. MR: 0275998
Flanders, H. (1974) A new proof of R. Foster's averaging formula in networks. Linear Algebra and Appl., 8, 35–37. MR: 0329772
Fleischmann, K. and Siegmund-Schultze, R. (1977) The structure of reduced critical Galton-Watson processes. Math. Nachr., 79, 233–241. MR: 0461689
Foguel, S.R (1971) On the “zero-two” law. Israel J. Math., 10, 275–280. MR: 0298759
Foguel, S.R (1976) More on the “zero-two” law. Proc. Amer. Math. Soc., 61(2), 262–264 (1977). MR: 0428076
Følner, E. (1955) On groups with full Banach mean value. Math. Scand., 3, 243–254. MR: 18,51f
Fontes, L.R, Schonmann, R.H., and Sidoravicius, V. (2002) Stretched exponential fixation in stochastic Ising models at zero temperature. Comm. Math. Phys., 228(3), 495–518. MR: 1918786
Ford, L.RJr. and Fulkerson, D.R. (1962) Flows in Networks. Princeton University Press, Princeton, NJ. MR: 28:2917
Fortuin, C.M (1972a) On the random-cluster model. II. The percolation model. Physica, 58, 393–418. MR: 51:14826
Fortuin, C.M (1972b) On the random-cluster model. III. The simple random-cluster model. Physica, 59, 545–570. MR: 55:5127
Fortuin, C.M and Kasteleyn, P.W. (1972) On the random-cluster model. I. Introduction and relation to other models. Physica, 57, 536–564. MR: 0359655
Fortuin, C.M, Kasteleyn, P.W., and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Comm. Math. Phys., 22, 89–103. MR: 46:8607
Foster, R.M (1948) The average impedance of an electrical network. In Reissner Anniversary Volume, Contributions to Applied Mechanics, pages 333–340. J. W. Edwards, Ann Arbor, Michigan. Edited by the Staff of the Department of Aeronautical Engineering and Applied Mechanics of the Polytechnic Institute of Brooklyn. MR: 10,662a
Frostman, O. (1935) Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddel. Lunds Univ. Mat. Sem., 3, 1–118.
Fukushima, M. (1980) Dirichlet Forms and Markov Processes. North-Holland, Amsterdam. MR: 81f:60105
Fukushima, M. (1985) Energy forms and diffusion processes. In Streit, L., editor, Mathematics + Physics. Vol. 1, pages 65–97. World Scientific, Singapore. Lectures on recent results. MR: 87m:60176
Furstenberg, H. (1963) A Poisson formula for semi-simple Lie groups. Ann. of Math. (2), 77, 335–386. MR: 0146298
Furstenberg, H. (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1, 1–49. MR: 35:4369
Furstenberg, H. (1970) Intersections of Cantor sets and transversality of semigroups. In Gunning, R.C., editor, Problems in Analysis, pages 41–59. Princeton University Press, Princeton, NJ. A symposium in honor of Salomon Bochner, Princeton University, Princeton, NJ, 1–3 April 1969. MR: 50:7040
Furstenberg, H. (1971a) Boundaries of Lie groups and discrete subgroups. In Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pages 301–306. Gauthier-Villars, Paris. MR: 0430160
Furstenberg, H. (1971b) Random walks and discrete subgroups of Lie groups. In Advances in Probability and Related Topics, Vol. 1, pages 1–63. Dekker, New York. MR: 0284569
Furstenberg, H. (1973) Boundary theory and stochastic processes on homogeneous spaces. In Moore, C.C., editor, Harmonic Analysis on Homogeneous Spaces, pages 193–229. Amer. Math. Soc., Providence R.I. Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972). MR: 0352328
Furstenberg, H. (2008) Ergodic fractal measures and dimension conservation. Ergodic Theory Dynam. Systems, 28(2), 405–422. MR: 2408385
Furstenberg, H. and Weiss, B. (2003) Markov processes and Ramsey theory for trees. Combin. Probab. Comput., 12(5–6), 547–563. MR: 2037069
Gaboriau, D. (1998) Mercuriale de groupes et de relations. C. R. Acad. Sci. Paris Sér. I Math., 326(2), 219–222. MR: 99h:28034
Gaboriau, D. (2000) Coût des relations d'équivalence et des groupes. Invent. Math., 139(1), 41–98. MR: 1728 876
Gaboriau, D. (2002) Invariants l2 de relations d'équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci., 95, 93–150. MR: 1953 191
Gaboriau, D. (2005) Invariant percolation and harmonic Dirichlet functions. Geom. Funct. Anal., 15(5), 1004–1051. MR: 2221157
Gaboriau, D. and Lyons, R. (2009) A measurable-group-theoretic solution to von Neumann's problem. Invent. Math., 177(3), 533–540. MR: 2534099
Gaboriau, D. and Tucker-Drob, R. (2015) Approximations of standard equivalence relations and Bernoulli percolation at pu. Preprint,
Gandolfi, A., Keane, M.S., and Newman, C.M. (1992) Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields, 92(4), 511–527. MR: 93f:60149
Gantert, N., Müller, S., Popov, S., and Vachkovskaia, M. (2012) Random walks on Galton-Watson trees with random conductances. Stochastic Process. Appl., 122(4), 1652–1671. MR: 2914767
Garban, C., Pete, G., and Schramm, O. (2013) The scaling limits of the minimal spanning tree and invasion percolation in the plane. Preprint,
Garnett, J.B and Marshall, D.E. (2005) Harmonic Measure. Vol. 2 of New Mathematical Monographs. Cambridge University Press, Cambridge. MR: 2150803
Gaudillière, A. and Landim, C. (2014) A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Related Fields, 158(1–2), 55–89. MR: 3152780
Gautero, F. and Mathéus, F. (2012) Poisson boundary of groups acting on.-trees. Israel J. Math., 191(2), 585–646. MR: 3011489
Geiger, J. (1995) Contour processes of random trees. In Etheridge, A., editor, Stochastic Partial Differential Equations, vol. 216 of London Math. Soc. Lecture Note Ser., pages 72–96. Cambridge University Press, Cambridge. Papers from the workshop held at the University of Edinburgh, Edinburgh, March 1994. MR: 1352736
Geiger, J. (1999) Elementary new proofs of classical limit theorems for Galton-Watson processes. J. Appl. Probab., 36(2), 301–309. MR: 1724 856
Georgakopoulos, A. (2010) Lamplighter graphs do not admit harmonic functions of finite energy. Proc. Amer. Math. Soc., 138(9), 3057–3061. MR: 2653930
Georgakopoulos, A. (2016) The boundary of a square tiling of a graph coincides with the Poisson boundary. Invent. Math., 203(3), 773–821. MR: 3461366
Georgakopoulos, A. and Winkler, P. (2014) New bounds for edge-cover by random walk. Combin. Probab. Comput., 23(4), 571–584. MR: 3217361
Gerl, P. (1988) Random walks on graphs with a strong isoperimetric property. J. Theoret. Probab., 1(2), 171–187. MR: 89g:60216
Gilch, L.A (2007) Rate of escape of random walks on free products. J. Aust. Math. Soc., 83(1), 31–54. MR: 2378433
Gilch, L.A (2011) Asymptotic entropy of random walks on free products. Electron. J. Probab., 16, paper no. 3, 76–105. MR: 2749773
Glasner, E. and Weiss, B. (1997) Kazhdan's property T and the geometry of the collection of invariant measures. Geom. Funct. Anal., 7(5), 917–935. MR: 99f:28029
Glasser, M.L and Zucker, I.J. (1977) Extended Watson integrals for the cubic lattices. Proc. Nat. Acad. Sci. U.S.A., 74(5), 1800–1801. MR: 0442300
Gneiting, T. and Raftery, A.E. (2007) Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc., 102(477), 359–378. MR: 2345548
Godsil, C.D, Imrich, W., Seifter, N., Watkins, M.E., and Woess, W. (1989) A note on bounded automorphisms of infinite graphs. Graphs Combin., 5(4), 333–338. MR: 1032384
Goldman, J.R and Kauffman, L.H. (1993) Knots, tangles, and electrical networks. Adv. in Appl. Math., 14(3), 267–306. MR: 94m:57013
Gouëzel, S., Mathéus, F., and Maucourant, F. (2015) Sharp lower bounds for the asymptotic entropy of symmetric random walks. Groups Geom. Dyn., 9(3), 711–735. MR: 3420541
Grabowski, L. (2014) On Turing dynamical systems and the Atiyah problem. Invent. Math., 198(1), 27–69. MR: 3260857
Graf, S. (1987) Statistically self-similar fractals. Probab. Theory Related Fields, 74(3), 357–392. MR: 88c:60038
Graf, S., Mauldin, R.D., and Williams, S.C. (1988) The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc., 71(381), x+121. MR: 88k:28010
Gravner, J. and Holroyd, A.E. (2009) Local bootstrap percolation. Electron. J. Probab., 14, paper no. 14, 385–399. MR: 2480546
Grey, D.R (1980) A new look at convergence of branching processes. Ann. Probab., 8(2), 377–380. MR: 81e:60091
Griffeath, D. and Liggett, T.M. (1982) Critical phenomena for Spitzer's reversible nearest particle systems. Ann. Probab., 10(4), 881–895. MR: 84f:60140
Grigorchuk, R.I (1980) Symmetrical random walks on discrete groups. In Dobrushin, R.L., Sinai, Ya.G., and Griffeath, D., editors, Multicomponent Random Systems, vol. 6 of Adv. Probab. Related Topics, pages 285–325. Dekker, New York. Translated from the Russian. MR: 599539
Grigorchuk, R.I (1983) On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR, 271(1), 30–33. MR: 85g:20042
Grigor'yan, A.A (1985) The existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds. Mat. Sb. (N.S.), 128(170)(3), 354–363, 446. English translation: Math. USSR-Sb. 56 (1987), no. 2, 349–358. MR: 87d:58140
Grimmett, G. (1999) Percolation, 2nd ed. Springer-Verlag, Berlin. MR: 1707 339
Grimmett, G. (2006) The Random-Cluster Model. Vol. 333 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. MR: 2243761
Grimmett, G. and Kesten, H. (1983) Random electrical networks on complete graphs II: Proofs. Unpublished manuscript, available at
Grimmett, G.R, Kesten, H., and Zhang, Y. (1993) Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields, 96(1), 33–44. MR: 94i:60078
Grimmett, G.R and Marstrand, J.M. (1990) The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A, 430(1879), 439–457. MR: 91m:60186
Grimmett, G.R and Newman, C.M. (1990) Percolation in 1 + 1 dimensions. In Grimmett, G.R. and Welsh, D.J.A., editors, Disorder in Physical Systems, pages 167–190. Oxford University Press, New York. A volume in honour of John M. Hammersley on the occasion of his 70th birthday. MR: 92a:60207
Grimmett, G.R and Stacey, A.M. (1998) Critical probabilities for site and bond percolation models. Ann. Probab., 26(4), 1788–1812. MR: 1675079
Grimmett, G.R and Stirzaker, D.R. (2001) Probability and Random Processes, 3rd ed. Oxford University Press, New York. MR: 2059709
Gromov, M. (1981a) Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math., 53, 53–73. MR: 83b:53041
Gromov, M. (1981b) Structures métriques pour les variétés riemanniennes. Vol. 1 of Textes Mathématiques [Mathematical Texts]. CEDIC, Paris. Edited by J., Lafontaine and P., Pansu. MR: 682063
Gromov, M. (1983) Filling Riemannian manifolds. J. Differential Geom., 18(1), 1–147. MR: 697984
Gromov, M. (1999) Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston. Based on the 1981 French original [MR 85e:53051], with appendices by M. Katz, P. Pansu, and S. Semmes, translated from the French by Sean Michael Bates. MR: 2000d:53065
Guillotin-Plantard, N. (2005) Gillis's random walks on graphs. J. Appl. Probab., 42(1), 295–301. MR: 2144913
Guivarc'h, Y. (1980) Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire. In Journées sur les Marches Aléatoires, vol. 74 of Astérisque, pages 47–98, 3. Soc. Math. France, Paris, Paris. Held at Kleebach, March 5–10, 1979. MR: 588157
Guttmann, A.J and Bursill, R.J. (1990) Critical exponent for the loop erased self-avoiding walk by Monte Carlo methods. J. Stat. Phys., 59(1–2), 1–9.
Guttorp, P. (1991) Statistical Inference for Branching Processes. Wiley Series in Probability and Mathematical Statistics. John Wiley, New York. MR: 1254434
Häggström, O. (1994) Aspects of Spatial Random Processes. Ph.D. thesis, Göteborg, Sweden.
Häggström, O. (1995) Random-cluster measures and uniform spanning trees. Stochastic Process. Appl., 59(2), 267–275. MR: 97b:60170
Häggström, O. (1997) Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab., 25(3), 1423–1436. MR: 98f:60207
Häggström, O. (1998) Uniform and minimal essential spanning forests on trees. Random Structures Algorithms, 12(1), 27–50. MR: 99i:05186
Häggström, O. (2013) Two badly behaved percolation processes on a nonunimodular graph. J. Theoret. Probab., 26(4), 1165–1180. MR: 3119989
Häggström, O., Jonasson, J., and Lyons, R. (2002) Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab., 30(1), 443–473. MR: 2003e:60220
Häggström, O. and Peres, Y. (1999) Monotonicity of uniqueness for percolation on Cayley graphs: All infinite clusters are born simultane- ously. Probab. Theory Related Fields, 113(2), 273–285. MR: 1676835
Häggström, O., Peres, Y., and Schonmann, R.H. (1999) Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In Bramson, M. and Durrett, R., editors, Perplexing Problems in Probability, pages 69–90. Birkhäuser, Boston. Festschrift in honor of Harry Kesten. MR: 2000b:60003
Häggström, O., Schonmann, R.H., and Steif, J.E. (2000) The Ising model on diluted graphs and strong amenability. Ann. Probab., 28(3), 1111–1137. MR: 2001i:60169
Hammersley, J.M (1959) Bornes supérieures de la probabilité critique dans un processus de filtration. In Le Calcul des Probabilités et ses Applications. Paris, 15–20 juillet 1958, Colloques Internationaux du Centre National de la Recherche Scientifique, LXXXVII, pages 17–37. Centre National de la Recherche Scientifique, Paris. MR: 0105751
Hammersley, J.M (1961a) Comparison of atom and bond percolation processes. J. Math. Phys., 2, 728–733. MR: 0130722
Hammersley, J.M (1961b) The number of polygons on a lattice. Proc. Cambridge Philos. Soc., 57, 516–523. MR: 23:A814
Han, T.S (1978) Nonnegative entropy measures of multivariate symmetric correlations. Information and Control, 36(2), 133–156. MR: 0464499
Hara, T. and Slade, G. (1990) Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys., 128(2), 333–391. MR: 91a:82037
Hara, T. and Slade, G. (1992) The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys., 4(2), 235–327. MR: 93j:82033
Hara, T. and Slade, G. (1994) Mean-field behaviour and the lace expansion. In Grimmett, G., editor, Probability and Phase Transition, pages 87–122. Kluwer Academic, Dordrecht. Proceedings of the NATO Advanced Study Institute on Probability Theory of Spatial Disorder and Phase Transition held at the University of Cambridge, Cambridge, July 4–16, 1993. MR: 95d:82033
Harmer, G.P and Abbott, D. (1999) Parrondo's paradox. Statist. Sci., 14(2), 206–213. MR: 1722065
de la Harpe, P. and Valette, A. (1989) La propriété de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger). Astérisque, 175, 158. With an appendix by M. Burger. MR: 1023471
Harris, T.E (1952) First passage and recurrence distributions. Trans. Amer. Math. Soc., 73, 471–486. MR: 0052057
Harris, T.E (1960) A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc., 56, 13–20. MR: 22:6023
Hawkes, J. (1970/71) Some dimension theorems for the sample functions of stable processes. Indiana Univ. Math. J., 20, 733–738. MR: 45:1251
Hawkes, J. (1981) Trees generated by a simple branching process. J. London Math. Soc. (2), 24(2), 373–384. MR: 83b:60072
He, Z.X and Schramm, O. (1995) Hyperbolic and parabolic packings. Discrete Comput. Geom., 14(2), 123–149. MR: 1331923
Heathcote, C.R (1966) Corrections and comments on the paper “A branching process allowing immigration”. J. Roy. Statist. Soc. Ser. B, 28, 213–217. MR: 33:1896b
Heathcote, C.R, Seneta, E., and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Teor. Verojatnost. i Primenen., 12, 341–346. MR: 36:978
Hebisch, W. and Saloff-Coste, L. (1993) Gaussian estimates for Markov chains and random walks on groups. Ann. Probab., 21(2), 673–709. MR: 1217561
Heyde, C.C (1970) Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Statist., 41, 739–742. MR: 40:8136
Heyde, C.C and Seneta, E. (1977) I. J. Bienaymé. Statistical Theory Anticipated. Vol. 3 of Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York. MR: 57:2855
Higuchi, Y. and Shirai, T. (2003) Isoperimetric constants of regular planar graphs. Interdiscip. Inform. Sci., 9(2), 221–228. MR: 2038013
Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 58, 13–30. MR: 0144363
Hoffman, C., Holroyd, A.E., and Peres, Y. (2006) A stable marriage of Poisson and Lebesgue. Ann. Probab., 34(4), 1241–1272. MR: 2257646
Holopainen, I. and Soardi, P.M. (1997) p-Harmonic functions on graphs and manifolds. Manuscripta Math., 94(1), 95–110. MR: 99c:31017
Holroyd, A.E (2003) Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields, 125(2), 195–224. MR: 1961342
Holroyd, A.E, Levine, L., Mész áros, K., Peres, Y., Propp, J., and Wilson, D.B. (2008) Chip-firing and rotor-routing on directed graphs. In Sidoravicius, V. and Vares, M.E., editors, In and Out of Equilibrium. 2, vol. 60 of Progr. Probab., pages 331–364. Birkhäuser, Basel. Papers from the 10th Brazilian School of Probability (EBP) held in Rio de Janeiro, July 30–August 4, 2006. MR: 2477390
Hoory, S., Linial, N., and Wigderson, A. (2006) Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4), 439–561 (electronic). MR: 2247919
Horn, R.A and Johnson, C.R. (2013) Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge. MR: 2978290
Houdayer, C. (2012) Invariant percolation and measured theory of nonamenable groups [after Gaboriau-Lyons, Ioana, Epstein]. Astérisque, 348, Exp. No. 1039, ix, 339–374. Séminaire Bourbaki: Vol. 2010/2011. Exposés 1027–1042. MR: 3051202
Howard, C.D (2000) Zero-temperature Ising spin dynamics on the homogeneous tree of degree three. J. Appl. Probab., 37(3), 736–747. MR: 1782449
Hutchcroft, T. (2015a) Wired cycle-breaking dynamics for uniform spanning forests. Ann. Probab. To appear,
Hutchcroft, T. (2015b) Interlacements and the wired uniform spanning forest. Preprint,
Hutchcroft, T. (2016) Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters. C. R. Math. Acad. Sci. Paris, 354(9), 944–947. MR: 3535351
Hutchcroft, T. and Nachmias, A. (2015) Indistinguishability of trees in uniform spanning forests. Probab. Theory Related Fields. To appear.
Hutchcroft, T. and Nachmias, A. (2016) Uniform spanning forests of planar graphs. Preprint,
Hutchcroft, T. and Peres, Y. (2015) Boundaries of planar graphs: A unified approach. Preprint,
Ichihara, K. (1978) Some global properties of symmetric diffusion processes. Publ. Res. Inst. Math. Sci., 14(2), 441–486. MR: 80d:60099
Imrich, W. (1975) On Whitney's theorem on the unique embeddability of 3-connected planar graphs. In Fiedler, M., editor, Recent Advances in Graph Theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pages 303–306. (Loose errata). Academia, Prague. MR: 52:5462
Indyk, P. and Motwani, R. (1999) Approximate nearest neighbors: Towards removing the curse of dimensionality. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing held in Dallas, TX, May 23–26, 1998, pages 604–613. ACM, New York. MR: 1715608
Jackson, T.S and Read, N. (2010a) Theory of minimum spanning trees. I. Mean-field theory and strongly disordered spin-glass model. Phys. Rev. E, 81(2), 021130.
Jackson, T.S and Read, N. (2010b) Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold. Phys. Rev. E, 81(2), 021131.
Jain, N.C and Taylor, S.J. (1973) Local asymptotic laws for Brownian motion. Ann. Probab., 1, 527–549. MR: 0365732
James, N. and Peres, Y. (1996) Cutpoints and exchangeable events for random walks. Teor. Veroyatnost. i Primenen., 41(4), 854–868. MR: 1687097
Janson, S. (1997) Gaussian Hilbert Spaces. Vol. 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge. MR: 1474726
J árai, A.A. and Redig, F. (2008) Infinite volume limit of the abelian sandpile model in dimensions d3. Probab. Theory Related Fields, 141(1–2), 181–212. MR: 2372969
J árai, A.A. and Werning, N. (2014) Minimal configurations and sandpile measures. J. Theoret. Probab., 27(1), 153–167. MR: 3174221
Jerrum, M. and Sinclair, A. (1989) Approximating the permanent. SIAM J. Comput., 18(6), 1149–1178. MR: 1025467
Joag-Dev, K., Perlman, M.D., and Pitt, L.D. (1983) Association of normal random variables and Slepian's inequality. Ann. Probab., 11(2), 451–455. MR: 690142
Joag-Dev, K. and Proschan, F. (1983) Negative association of random variables, with applications. Ann. Statist., 11(1), 286–295. MR: 85d:62058
Joffe, A. and Waugh, W.A.O'N. (1982) Exact distributions of kin numbers in a Galton-Watson process. J. Appl. Probab., 19(4), 767–775. MR: 84a:60104
Johnson, W.B and Lindenstrauss, J. (1984) Extensions of Lipschitz mappings into a Hilbert space. In Beals, R., Beck, A., Bellow, A., and Hajian, A., editors, Conference in Modern Analysis and Probability, vol. 26 of Contemp. Math., pages 189–206. Amer. Math. Soc., Providence, RI. Held at Yale University, New Haven, Conn., June 8–11, 1982, Held in honor of Professor Shizuo Kakutani. MR: 737400
Jolissaint, P.N and Valette, A. (2014) Lp-distortion and p-spectral gap of finite graphs. Bull. Lond. Math. Soc., 46(2), 329–341. MR: 3194751
Jorgensen, P.ET.and Pearse, E.P.J. (2008) Operator Theory of Electrical Resistance Networks. Universitext. Springer-Verlag. To appear,
Joyal, A. (1981) Une théorie combinatoire des séries formelles. Adv. in Math., 42(1), 1–82. MR: 633783
Kahn, J. (2003) Inequality of two critical probabilities for percolation. Electron. Comm. Probab., 8, 184–187 (electronic). MR: 2042 758
Kahn, J., Kim, J.H., Lovász, L., and Vu, V.H. (2000) The cover time, the blanket time, and the Matthews bound. In 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000), pages 467–475. IEEE Comput. Soc. Press, Los Alamitos, CA. MR: 1931843
Kahn, J.D, Linial, N., Nisan, N., and Saks, M.E. (1989) On the cover time of random walks on graphs. J. Theoret. Probab., 2(1), 121–128. MR: 981769
Kaimanovich, V.A (1985) An entropy criterion of maximality for the boundary of random walks on discrete groups. Dokl. Akad. Nauk SSSR, 280(5), 1051–1054. MR: 780288
Kaimanovich, V.A (1990) Boundary and entropy of random walks in random environment. In Grigelionis, B., Sazonov, V.V., and Statulevicius, V., editors, Probability Theory and Mathematical Statistics. Vol. I, pages 573–579. “Mokslas,” Vilnius. Proceedings of the Fifth Conference held in Vilnius, June 25–July 1, 1989. MR: 1153846
Kaimanovich, V.A (1992) Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators. Potential Anal., 1(1), 61–82. MR: 94i:31012
Kaimanovich, V.A (1994) The Poisson boundary of hyperbolic groups. C. R. Acad. Sci. Paris Sér. I Math., 318(1), 59–64. MR: 1260536
Kaimanovich, V.A (1996) Boundaries of invariant Markov operators: The identification problem. In Pollicott, M. and Schmidt, K., editors, Ergodic Theory of Zd Actions, vol. 228 of London Math. Soc. Lecture Note Ser., pages 127–176. Cambridge University Press, Cambridge, Cambridge. Proceedings of the symposium held in Warwick, 1993–1994. MR: 1411218
Kaimanovich, V.A (2000) The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2), 152(3), 659–692. MR: 1815698
Kaimanovich, V.A (2001) Poisson boundary of discrete groups. Preprint,
Kaimanovich, V.A (2005) “Münchhausen trick” and amenability of self-similar groups. Internat. J. Algebra Comput., 15(5–6), 907–937. MR: 2197814
Kaimanovich, V.A and Masur, H. (1996) The Poisson boundary of the mapping class group. Invent. Math., 125(2), 221–264. MR: 1395719
Kaimanovich, V.A and Masur, H. (1998) The Poisson boundary of Teichmüller space. J. Funct. Anal., 156(2), 301–332. MR: 1636940
Kaimanovich, V.A and Vershik, A.M. (1983) Random walks on discrete groups: Boundary and entropy. Ann. Probab., 11(3), 457–490. MR: 85d:60024
Kaimanovich, V.A and Woess, W. (2002) Boundary and entropy of space homogeneous Markov chains. Ann. Probab., 30(1), 323–363. MR: 1894110
Kakutani, S. (1944) Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo, 20, 706–714. MR: 7,315b
Kamae, T. (1982) A simple proof of the ergodic theorem using nonstandard analysis. Israel J. Math., 42(4), 284–290. MR: 682311
Kanai, M. (1985) Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan, 37(3), 391–413. MR: 87d:53082
Kanai, M. (1986) Rough isometries and the parabolicity of Riemannian manifolds. J. Math. Soc. Japan, 38(2), 227–238. MR: 87e:53066
Kargapolov, M.I and Merzljakov, J.I. (1979) Fundamentals of the Theory of Groups. Springer-Verlag, New York. Translated from the second Russian edition by Robert G. Burns. MR: 80k:20002
Karlsson, A. (2003) Boundaries and random walks on finitely generated infinite groups. Ark. Mat., 41(2), 295–306. MR: 2011923
Karlsson, A. and Ledrappier, F. (2007) Linear drift and Poisson boundary for random walks. Pure Appl. Math. Q., 3(4, Special Issue: In honor of Grigory Margulis. Part 1), 1027–1036. MR: 2402595
Karlsson, A. and Woess, W. (2007) The Poisson boundary of lamplighter random walks on trees. Geom. Dedicata, 124, 95–107. MR: 2318539
Kassel, A. and Wilson, D.B. (2016) The looping rate and sandpile density of planar graphs. Amer. Math. Monthly, 123(1), 19–39. MR: 3453533
Kasteleyn, P. (1961) The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice. Physica, 27, 1209–1225.
Katznelson, Y. and Weiss, B. (1982) A simple proof of some ergodic theorems. Israel J. Math., 42(4), 291–296. MR: 682312
Kazami, T. and Uchiyama, K. (2008) Random walks on periodic graphs. Trans. Amer. Math. Soc., 360(11), 6065–6087. MR: 2425703
Keane, M. (1995) The essence of the law of large numbers. In Takahashi, Y., editor, Algorithms, Fractals, and Dynamics, pages 125–129. Plenum, New York. Papers from the Hayashibara Forum '92 International Symposium on New Bases for Engineering Science, Algorithms, Dynamics and Fractals held in Okayama, November 23–28, 1992, and the Symposium on Algorithms, Fractals and Dynamics held at Kyoto University, Kyoto, November 30–December 2, 1992. MR: 1402486
Kelly, J.B (1970) Metric inequalities and symmetric differences. In Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), pages 193–212. Academic Press, New York. MR: 0264600
Kemeny, J.G and Snell, J.L. (1960) Finite Markov Chains. The University Series in Undergraduate Mathematics. D. Van, Nostrand, Princeton, NJ. MR: 0115196
Kendall, D.G (1951) Some problems in the theory of queues. J. Roy. Statist. Soc. Ser. B., 13, 151–173; discussion: 173–185. MR: 0047944
Kennelly, A.E (1899) Equivalence of triangles and stars in conducting networks. Electrical World and Engineer, 34, 413–414.
Kenyon, R.W (1997) Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist., 33(5), 591–618. MR: 1473567
Kenyon, R.W (1998) Tilings and discrete Dirichlet problems. Israel J. Math., 105, 61–84. MR: 99m:52026
Kenyon, R.W (2000a) The asymptotic determinant of the discrete Laplacian. Acta Math., 185(2), 239–286. MR: 2002g:82019
Kenyon, R.W (2000b) Long-range properties of spanning trees. J. Math. Phys., 41(3), 1338–1363. MR: 1757 962
Kenyon, R.W (2001) Dominos and the Gaussian free field. Ann. Probab., 29(3), 1128–1137. MR: 1872739
Kenyon, R.W (2008) Height fluctuations in the honeycomb dimer model. Comm. Math. Phys., 281(3), 675–709. MR: 2415464
Kenyon, R.W, Propp, J.G., and Wilson, D.B. (2000) Trees and matchings. Electron. J. Combin., 7(1), Research Paper 25, 34 pp. (electronic). MR: 2001a:05123
Kenyon, R.W and Wilson, D.B. (2015) Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs. J. Amer. Math. Soc., 28(4), 985–1030. MR: 3369907
Kesten, H. (1959a) Full Banach mean values on countable groups. Math. Scand., 7, 146–156. MR: 22:2911
Kesten, H. (1959b) Symmetric random walks on groups. Trans. Amer. Math. Soc., 92, 336–354. MR: 22:253
Kesten, H. (1967) The Martin boundary of recurrent random walks on countable groups. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2, pages 51–74. University of California Press, Berkeley. MR: 0214137
Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals 12. Comm. Math. Phys., 74(1), 41–59. MR: 82c:60179
Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhäuser, Boston. MR: 84i:60145
Kesten, H. (1986) Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist., 22(4), 425–487. MR: 88b:60232
Kesten, H., Ney, P., and Spitzer, F. (1966) The Galton-Watson process with mean one and finite variance. Teor. Verojatnost. i Primenen., 11, 579–611. MR: 34:6868
Kesten, H. and Stigum, B.P. (1966) A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist., 37, 1211–1223. MR: 33:6707
Khoshnevisan, D., Peres, Y., and Xiao, Y. (2000) Limsup random fractals. Electron. J. Probab., 5, paper no. 5, 24 pp. MR: 1743726
Kifer, Y. (1986) Ergodic Theory of Random Transformations. Birkhäuser, Boston. MR: 89c:58069
Kingman, J.FC. (1968) The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B, 30, 499–510. MR: 0254907
Kirchhoff, G. (1847) Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. Chem., 72(12), 497–508.
Kleiner, B. (2010) A new proof of Gromov's theorem on groups of polynomial growth. J. Amer. Math. Soc., 23(3), 815–829. MR: 2629989
Kolmogorov, A. (1938) On the solution of a problem in biology. Izv. NII Matem. Mekh. Tomskogo Univ., 2, 7–12.
Kolmogorov, A.N and Barzdin', Y.M. (1967) On the realization of nets in 3-dimensional space. Probl. Cybernet, 19, 261–268. In Russian. See also Selected Works of A.N. Kolmogorov, Vol. III, pp. 194–202 (and a remark on p. 245), Kluwer Academic, 1993.
Korevaar, N.J and Schoen, R.M. (1997) Global existence theorems for harmonic maps to non-locally compact spaces. Comm. Anal. Geom., 5(2), 333–387. MR: 1483983
Kotani, M. and Sunada, T. (2000) Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo, 7(1), 7–25. MR: 1749978
Kozáková, I. (2008) Critical percolation of free product of groups. Internat. J. Algebra Comput., 18(4), 683–704. MR: 2428151
Kozdron, M.J, Richards, L.M., and Stroock, D.W. (2013) Determinants, their applications to Markov processes, and a random walk proof of Kirchhoff's matrix tree theorem. Preprint,
Kozma, G. (2011) Percolation on a product of two trees. Ann. Probab., 39(5), 1864–1895. MR: 2884876
Kozma, G. and Schreiber, E. (2004) An asymptotic expansion for the discrete harmonic potential. Electron. J. Probab., 9, paper no. 1, 1–17 (electronic). MR: 2041826
Krikun, M. (2007) Connected allocation to Poisson points in.2. Electron. Comm. Probab., 12, 140–145 (electronic). MR: 2318161
Kuipers, L. and Niederreiter, H. (1974) Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley-Interscience, New York. MR: 0419394
Kuratowski, K. (1966) Topology. Vol. I, revised and augmented ed. Academic Press, New York. Translated from the French by J. Jaworowski. MR: 36:840
Lalley, S.P (1998) Percolation on Fuchsian groups. Ann. Inst. H. Poincaré Probab. Statist., 34(2), 151–177. MR: 1614583
Lamperti, J. (1967) The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 7, 271–288. MR: 0217893
Lawler, G.F (1980) A self-avoiding random walk. Duke Math. J., 47(3), 655–693. MR: 81j:60081
Lawler, G.F (1983) A connective constant for loop-erased self-avoiding random walk. J. Appl. Probab., 20(2), 264–276. MR: 84g:60113
Lawler, G.F (1986) Gaussian behavior of loop-erased self-avoiding random walk in four dimensions. Duke Math. J., 53(1), 249–269. MR: 87i:60078
Lawler, G.F (1988) Loop-erased self-avoiding random walk in two and three dimensions. J. Statist. Phys., 50(1–2), 91–108. MR: 89f:82053
Lawler, G.F (1991) Intersections of Random Walks. Birkhäuser, Boston. MR: 92f:60122
Lawler, G.F (2014) The probability that planar loop-erased random walk uses a given edge. Electron. Commun. Probab., 19, paper no. 51, 13. MR: 3246970
Lawler, G.F, Schramm, O., and Werner, W. (2004a) Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32(1B), 939–995. MR: 2044 671
Lawler, G.F, Schramm, O., and Werner, W. (2004b) On the scaling limit of planar self-avoiding walk. In Lapidus, M.L. and van Frankenhuijsen, M., editors, Fractal Geometry and Applications: A Jubilee of Benoıt Mandelbrot. Part 2, vol. 72 of Proc. Sympos. Pure Math., pages 339–364. Amer. Math. Soc., Providence, RI. Proceedings of a Special Session of the Annual Meeting of the American Mathematical Society held in San Diego, CA, January 2002. MR: 2112127
Lawler, G.F and Sokal, A.D. (1988) Bounds on the L2 spectrum for Markov chains and Markov processes: A generalization of Cheeger's inequality. Trans. Amer. Math. Soc., 309(2), 557–580. MR: 89h:60105
Lawrencenko, S., Plummer, M.D., and Zha, X. (2002) Isoperimetric constants of infinite plane graphs. Discrete Comput. Geom., 28(3), 313–330. MR: 1923 955
Le Gall, J.F. (1987) Le comportement du mouvement brownien entre les deux instants ou il passe par un point double. J. Funct. Anal., 71(2), 246–262. MR: 880979
Le Gall, J.F. and Le Jan, Y. (1998) Branching processes in Lévy processes: The exploration process. Ann. Probab., 26(1), 213–252. MR: 1617047
Le Gall, J.F. and Rosen, J. (1991) The range of stable random walks. Ann. Probab., 19(2), 650–705. MR: 1106281
Ledrappier, F. (1984) Frontiere de Poisson pour les groupes discrets de matrices. C. R. Acad. Sci. Paris Sér. I Math., 298(16), 393–396. MR: 748930
Ledrappier, F. (1985) Poisson boundaries of discrete groups of matrices. Israel J. Math., 50(4), 319–336. MR: 800190
Ledrappier, F. (1992) Sharp estimates for the entropy. In Picardello, M.A., editor, Proceedings of the International Meeting held in Frascati, July 1–10, 1991, pages 281–288. Plenum, New York. MR: 1222466
Lee, J.R and Peres, Y. (2013) Harmonic maps on amenable groups and a diffusive lower bound for random walks. Ann. Probab., 41(5), 3392–3419. MR: 3127886
Lee, J.R, Peres, Y., and Smart, C.K. (2014) A Gaussian upper bound for martingale small-ball probabilities. Ann. Probab. To appear,
Lee, J.R and Qin, T. (2012) A note on mixing times of planar random walks. Unpublished manuscript,
Levin, D.A and Peres, Y. (2010) Pólya's theorem on random walks via Pólya's urn. Amer. Math. Monthly, 117(3), 220–231. MR: 2640849
Levin, D.A, Peres, Y., and Wilmer, E.L. (2009) Markov Chains and Mixing Times. American Mathematical Society, Providence, RI. With a chapter by James G., Propp and David B., Wilson. MR: 2466937
Levitt, G. (1995) On the cost of generating an equivalence relation. Ergodic Theory Dynam. Systems, 15(6), 1173–1181. MR: 96i:58091
Liggett, T.M (1985) An improved subadditive ergodic theorem. Ann. Probab., 13(4), 1279–1285. MR: 806224
Liggett, T.M (1987) Reversible growth models on symmetric sets. In Ito, K. and Ikeda, N., editors, Probabilistic Methods in Mathematical Physics, pages 275–301. Academic Press, Boston. Proceedings of the Taniguchi International Symposium held in Katata, June 20–26, 1985, and at Kyoto University, Kyoto, June 27–29, 1985. MR: 933828
Liggett, T.M, Schonmann, R.H., and Stacey, A.M. (1997) Domination by product measures. Ann. Probab., 25(1), 71–95. MR: 98f:60095
Lindvall, T. and Rogers, L.C.G. (1996) On coupling of random walks and renewal processes. J. Appl. Probab., 33(1), 122–126. MR: 1371959
Linial, N., London, E., and Rabinovich, Y. (1995) The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2), 215–245. MR: 1337355
Linial, N., Magen, A., and Naor, A. (2002) Girth and Euclidean distortion. Geom. Funct. Anal., 12(2), 380–394. MR: 1911665
Loomis, L.H and Whitney, H. (1949) An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc, 55, 961–962. MR: 0031538
Louder, L. and Souto, J. (2012) Diameter and spectral gap for planar graphs. Preprint,
Lovász, L. and Kannan, R. (1999) Faster mixing via average conductance. In Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999), pages 282–287. ACM, New York. MR: 1798047
Lubetzky, E. and Peres, Y. (2015) Cutoff on all Ramanujan graphs. Preprint,
Lubotzky, A., Phillips, R., and Sarnak, P. (1988) Ramanujan graphs. Combinatorica, 8(3), 261–277. MR: 963118
Lück, W. (2009) L2-invariants from the algebraic point of view. In Bridson, M.R., Kropholler, P.H., and Leary, I.J., editors, Geometric and Cohomological Methods in Group Theory, vol. 358 of London Math. Soc. Lecture Note Ser., pages 63–161. Cambridge University Press, Cambridge. Papers from the London Mathematical Society Symposium on Geometry and Cohomology in Group Theory held in Durham, July 2003. MR: 2605176
Lyons, R. (1988) Strong laws of large numbers for weakly correlated random variables. Michigan Math. J., 35(3), 353–359. MR: 90d:60038
Lyons, R. (1989) The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys., 125(2), 337–353. MR: 90h:82046
Lyons, R. (1990) Random walks and percolation on trees. Ann. Probab., 18(3), 931–958. MR: 91i:60179
Lyons, R. (1992) Random walks, capacity and percolation on trees. Ann. Probab., 20(4), 2043–2088. MR: 93k:60175
Lyons, R. (1995) Random walks and the growth of groups. C. R. Acad. Sci. Paris Sér. I Math., 320(11), 1361–1366. MR: 96e:60015
Lyons, R. (1996) Diffusions and random shadows in negatively curved manifolds. J. Funct. Anal., 138(2), 426–448. MR: 97d:58205
Lyons, R. (2000) Phase transitions on nonamenable graphs. J. Math. Phys., 41(3), 1099–1126. MR: 2001c:82028
Lyons, R. (2003) Determinantal probabilitymeasures. Publ.Math. Inst. Hautes Études Sci., 98(1), 167–212. MR: 2031202
Lyons, R. (2005) Asymptotic enumeration of spanning trees. Combin. Probab. Comput., 14(4), 491–522. MR: 2160416
Lyons, R. (2009) Random complexes and l2-Betti numbers. J. Topol. Anal., 1(2), 153–175. MR: 2541759
Lyons, R. (2010) Identities and inequalities for tree entropy. Combin. Probab. Comput., 19(2), 303–313. MR: 2593624
Lyons, R. (2013a) Distance covariance in metric spaces. Ann. Probab., 41(5), 3284–3305. MR: 3127883
Lyons, R. (2013b) Fixed price of groups and percolation. Ergodic Theory Dynam. Systems, 33(1), 183–185. MR: 3009109
Lyons, R., Morris, B.J., and Schramm, O. (2008) Ends in uniform spanning forests. Electron. J. Probab., 13, paper no. 58, 1702–1725. MR: 2448128
Lyons, R. and Nazarov, F. (2011) Perfect matchings as IID factors on non-amenable groups. European J. Combin., 32(7), 1115–1125. MR: 2825538
Lyons, R. and Pemantle, R. (1992) Random walk in a random environment and first-passage percolation on trees. Ann. Probab., 20(1), 125–136. MR: 93c:60103
Lyons, R. and Pemantle, R. (2003) Correction: “Random walk in a random environment and first-passage percolation on trees” [Ann. Probab. 20(1992) no. 1, 125–136; MR 93c:60103]. Ann. Probab., 31(1), 528–529. MR: 1959 801
Lyons, R., Pemantle, R., and Peres, Y. (1995a) Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab., 23(3), 1125–1138. MR: 96m:60194
Lyons, R., Pemantle, R., and Peres, Y. (1995b) Ergodic theory on Galton-Watson trees: Speed of random walk and dimension of harmonic measure. Ergodic Theory Dynam. Systems, 15(3), 593–619. MR: 96e:60125
Lyons, R., Pemantle, R., and Peres, Y. (1996a) Biased random walks on Galton-Watson trees. Probab. Theory Related Fields, 106(2), 249–264. MR: 97h:60094
Lyons, R., Pemantle, R., and Peres, Y. (1996b) Random walks on the lamplighter group. Ann. Probab., 24(4), 1993–2006. MR: 97j:60014
Lyons, R., Pemantle, R., and Peres, Y. (1997) Unsolved problems concerning random walks on trees. In Athreya, K.B. and Jagers, P., editors, Classical and Modern Branching Processes, pages 223–237. Springer, New York. Papers from the IMAWorkshop held at the University of Minnesota, Minneapolis, MN, June 13–17, 1994. MR: 98j:60098
Lyons, R. and Peres, Y. (2015a) Cycle density in infinite Ramanujan graphs. Ann. Probab., 43(6), 3337–3358. MR: 3433583
Lyons, R. and Peres, Y. (2015b) Poisson boundaries of lamplighter groups: Proof of the Kaimanovich-Vershik conjecture. Preprint,
Lyons, R., Peres, Y., and Schramm, O. (2003) Markov chain intersections and the loop-erased walk. Ann. Inst. H. Poincaré Probab. Statist., 39(5), 779–791. MR: 1997 212
Lyons, R., Peres, Y., and Schramm, O. (2006) Minimal spanning forests. Ann. Probab., 34(5), 1665–1692. MR: 2271476
Lyons, R., Pichot, M., and Vassout, S. (2008) Uniform non-amenability, cost, and the first l2-Betti number. Groups Geom. Dyn., 2(4), 595–617. MR: 2442947
Lyons, R. and Schramm, O. (1999) Indistinguishability of percolation clusters. Ann. Probab., 27(4), 1809–1836. MR: 1742 889
Lyons, R. and Steif, J.E. (2003) Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J., 120(3), 515–575. MR: 2030095
Lyons, R. and Thom, A. (2016) Invariant coupling of determinantal measures on sofic groups. Ergodic Theory Dynam. Systems, 36(2), 574–607.
Lyons, T. (1983) A simple criterion for transience of a reversible Markov chain. Ann. Probab., 11(2), 393–402. MR: 84e:60102
Lyons, T. (1987) Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differential Geom., 26(1), 33–66. MR: 892030
Lyons, T.J and Zhang, T.S. (1994) Decomposition of Dirichlet processes and its application. Ann. Probab., 22(1), 494–524. MR: 1258888
Lyons, T.J and Zheng, W.A. (1988) A crossing estimate for the canonical process on a Dirichlet space and a tightness result. Astérisque, 157–158, 249–271. Papers from the colloquium held in Palaiseau, June 22–26, 1987. MR: 976222
Maak, W. (1935) Eine neue Definition der fastperiodischen Funktionen. Abh. Math. Semin. Hamb. Univ., 11, 240–244.
Macpherson, H.D (1982) Infinite distance transitive graphs of finite valency. Combinatorica, 2(1), 63–69. MR: 671146
Mader, W. (1970) Ü ber den Zusammenhang symmetrischer Graphen. Arch. Math. (Basel), 21, 331–336. MR: 44:6534
Madras, N. and Slade, G. (1993) The Self-Avoiding Walk. Birkhäuser, Boston. MR: 94f:82002
Maher, J. and Tiozzo, G. (2014) Random walks on weakly hyperbolic groups. J. Reine Angew. Math. To appear,
Mairesse, J. and Mathéus, F. (2007a) Random walks on free products of cyclic groups. J. Lond. Math. Soc. (2), 75(1), 47–66. MR: 2302729
Mairesse, J. and Mathéus, F. (2007b) Randomly growing braid on three strands and the manta ray. Ann. Appl. Probab., 17(2), 502–536. MR: 2308334
Majumdar, S.N (1992) Exact fractal dimension of the loop-erased self-avoiding random walk in two dimensions. Phys. Rev. Lett., 68, 2329–2331.
Malyutin, A.V (2003) The Poisson-Furstenberg boundary of a locally free group. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 301(Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 9), 195–211, 245. MR: 2032055
Malyutin, A.V, Nagnibeda, T., and Serbin, D. (2016) Boundaries of.n-free groups. In Ceccherini-Silberstein, T. et al., editors, Groups, Graphs, and Random Walks, London Math. Soc. Lecture Notes Ser. Cambridge University Press. To appear,
Malyutin, A.V and Svetlov, P. (2014) Poisson-Furstenberg boundaries of fundamental groups of closed 3-manifolds. Preprint,
Mandelbrot, B.B (1982) The Fractal Geometry of Nature. W. H. Freeman, San Francisco. Schriftenreihe für den Referenten [Series for the Referee]. MR: 665254
Mann, A. (2012) How Groups Grow. Vol. 395 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge. MR: 2894945
Marchal, P. (1998) The best bounds in a theorem of Russell Lyons. Electron. Comm. Probab., 3, paper no. 11, 91–94 (electronic). MR: 1650563
Marchal, P. (2000) Loop-erased random walks, spanning trees and Hamiltonian cycles. Electron. Comm. Probab., 5, 39–50 (electronic). MR: 1736723
Marckert, J.F (2008) The lineage process in Galton-Watson trees and globally centered discrete snakes. Ann. Appl. Probab., 18(1), 209–244. MR: 2380897
Marckert, J.F and Mokkadem, A. (2003) Ladder variables, internal structure of Galton-Watson trees and finite branching random walks. J. Appl. Probab., 40(3), 671–689. MR: 1993260
Margulis, G.A (1988) Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii, 24(1), 51–60. MR: 939574
Martineau, S. and Tassion, V. (2013) Locality of percolation for abelian Cayley graphs. Ann. Probab. To appear,
Massey, W.S (1991) A Basic Course in Algebraic Topology. Springer-Verlag, New York. MR: 92c:55001
Masson, R. (2009) The growth exponent for planar loop-erased random walk. Electron. J. Probab., 14, paper no. 36, 1012–1073. MR: 2506124
Matousek, J. (2002) Lectures on Discrete Geometry. Vol. 212 of Graduate Texts in Mathematics. Springer-Verlag, New York. MR: 1899299
Matousek, J. (2008) On variants of the Johnson-Lindenstrauss lemma. Random Structures Algorithms, 33(2), 142–156. MR: 2436844
Matsuzaki, K. and Taniguchi, M. (1998) Hyperbolic Manifolds and Kleinian Groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York. MR: 1638795
Matthews, P. (1988) Covering problems for Brownian motion on spheres. Ann. Probab., 16(1), 189–199. MR: 920264
Mattila, P. (1995) Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Vol. 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. MR: 1333890
Mauldin, R.D and Williams, S.C. (1986) Random recursive constructions: Asymptotic geometric and topological properties. Trans. Amer. Math. Soc., 295(1), 325–346. MR: 87j:60027
McCrea, W.H and Whipple, F.J.W. (1940) Random paths in two and three dimensions. Proc. Roy. Soc. Edinburgh, 60, 281–298. MR: 2,107f
McDiarmid, C. (1989) On the method of bounded differences. In Siemons, J., editor, Surveys in Combinatorics, 1989, vol. 141 of London Math. Soc. Lecture Note Ser., pages 148–188. Cambridge University Press, Cambridge. Papers from the Twelfth British Combinatorial Conference held at the University of East Anglia, Norwich, 1989. MR: 1036755
McGuinness, S. (1989) Random Walks on Graphs and Directed Graphs. Ph.D. thesis, University of Waterloo.
Medolla, G. and Soardi, P.M. (1995) Extension of Foster's averaging formula to infinite networks with moderate growth. Math. Z., 219(2), 171–185. MR: 96g:94031
Meir, A. and Moon, J.W. (1970) The distance between points in random trees. J. Combinatorial Theory, 8, 99–103. MR: 0263685
Merkl, F. and Rolles, S.W.W. (2009) Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab., 37(5), 1679– 1714. MR: 2561431
Mester, P. (2013) Invariant monotone coupling need not exist. Ann. Probab., 41(3A), 1180–1190. MR: 3098675
Minc, H. (1988) Nonnegative Matrices.Wiley-Interscience Series in Discrete Mathematics and Optimization. JohnWiley, New York. MR: 89i:15001
Mohar, B. (1988) Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra Appl., 103, 119–131. MR: 89k:05071
Mok, N. (1995) Harmonic forms with values in locally constant Hilbert bundles. J. Fourier Anal. Appl., Special Issue, 433–453. Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). MR: 1364901
Montroll, E. (1964) Lattice statistics. In Beckenbach, E., editor, Applied Combinatorial Mathematics, pages 96–143. John Wiley, New York. University of California Engineering and Physical Sciences Extension Series. MR: 30:4687
Moon, J.W (1967) Various proofs of Cayley's formula for counting trees. In Harary, F. and Beineke, L., editors, A Seminar on Graph Theory, pages 70–78. Holt, Rinehart and Winston, New York. MR: 35:5365
Mori, A. (1954) A note on unramified abelian covering surfaces of a closed Riemann surface. J. Math. Soc. Japan, 6, 162–176. MR: 0066468
Morris, B. (2003) The components of the wired spanning forest are recurrent. Probab. Theory Related Fields, 125(2), 259–265. MR: 1961 344
Morris, B. and Peres, Y. (2005) Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields, 133(2), 245–266. MR: 2198701
Mörters, P. and Peres, Y. (2010) Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. With an appendix by Oded Schramm and Wendelin Werner. MR: 2604525
Muchnik, R. and Pak, I. (2001) Percolation on Grigorchuk groups. Comm. Algebra, 29(2), 661–671. MR: 2002e:82033
Murty, M.R (2003) Ramanujan graphs. J. Ramanujan Math. Soc., 18(1), 33–52. MR: 1966527
Nachbin, L. (1965) The Haar Integral. D. Van Nostrand, Princeton, NJ. MR: 0175995
Naor, A. (2010) L1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry. In Bhatia, R., Pal, A., Rangarajan, G., Srinivas, V., and Vanninathan, M., editors, Proceedings of the International Congress of Mathematicians. Volume III, pages 1549–1575. Hindustan Book Agency, New Delhi. MR: 2827855
Naor, A. and Peres, Y. (2008) Embeddings of discrete groups and the speed of random walks. Int. Math. Res. Not. IMRN, 2008, Art. rnn 076, 34. MR: 2439557
Naor, A., Peres, Y., Schramm, O., and Sheffield, S. (2006) Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1), 165–197. MR: 2239346
Nash-Williams, C.St.J.A. (1959) Random walk and electric currents in networks. Proc. Cambridge Philos. Soc., 55, 181–194. MR: 23:A2239
Nash-Williams, C.St.J.A. (1961) Edge-disjoint spanning trees of finite graphs. J. London Math. Soc., 36, 445–450. MR: 0133253
Neumann, M. and Sze, N.S. (2011) On the inverse mean first passage matrix problem and the inverse M-matrix problem. Linear Algebra Appl., 434(7), 1620–1630. MR: 2775741
Neveu, J. (1986) Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist., 22(2), 199–207. MR: 88a:60150
Nevo, A. and Sageev, M. (2013) The Poisson boundary of CAT??0_ cube complex groups. Groups Geom. Dyn., 7(3), 653–695. MR: 3095714
Newman, C.M (1997) Topics in Disordered Systems. Birkhäuser, Basel. MR: 99e:82052
Newman, C.M and Schulman, L.S. (1981) Infinite clusters in percolation models. J. Statist. Phys., 26(3), 613–628. MR: 83e:82038
Newman, C.M and Stein, D.L. (1996) Ground-state structure in a highly disordered spin-glass model. J. Statist. Phys., 82(3–4), 1113–1132. MR: 97a:82054
Newman, C.M and Stein, D.L. (2006) Short-range spin glasses: Selected open problems. In Bovier, A., Dunlop, F., van Enter, A., den Hollander, F., and Dalibard, J., editors, Mathematical Statistical Physics, pages 273–293. Elsevier, Amsterdam. Papers from the 83rd Session of the Summer School in Physics held in Les Houches, July 4–29, 2005. MR: 2581887
Nienhuis, B. (1982) Exact critical point and critical exponents of On models in two dimensions. Phys. Rev. Lett., 49(15), 1062–1065.
Nilli, A. (1991) On the second eigenvalue of a graph. Discrete Math., 91(2), 207–210. MR: 1124768
Nilli, A. (2004) Tight estimates for eigenvalues of regular graphs. Electron. J. Combin., 11(1), Note 9, 4 pp. (electronic). MR: 2056091
Norris, J., Peres, Y., and Zhai, A. (2015) Surprise probabilities in Markov chains. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 1759–1773. Society for Industrial and Applied Mathematics (SIAM), Philadelphia.
Northshield, S. (1992) Cogrowth of regular graphs. Proc. Amer. Math. Soc., 116(1), 203–205. MR: 1120509
Ornstein, D.S and Weiss, B. (1987) Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math., 48, 1–141. MR: 88j:28014
Orstein, D.S and Sucheston, L. (1970) An operator theorem on L1 convergence to zero with applications to Markov kernels. Ann. Math. Statist., 41, 1631–1639. MR: 0272057
Ozawa, N. (2015) A functional analysis proof of Gromov's polynomial growth theorem. Preprint,
Pak, I. and Smirnova-Nagnibeda, T. (2000) On non-uniqueness of percolation on nonamenable Cayley graphs. C. R. Acad. Sci. Paris Sér. I Math., 330(6), 495–500. MR: 1756 965
Pakes, A.G and Dekking, F.M. (1991) On family trees and subtrees of simple branching processes. J. Theoret. Probab., 4(2), 353–369. MR: 92f:60145
Pakes, A.G and Khattree, R. (1992) Length-biasing, characterizations of laws and the moment problem. Austral. J. Statist., 34(2), 307–322. MR: 94a:60018
Paley, R.EA.C. and Zygmund, A. (1932) A note on analytic functions in the unit circle. Proc. Camb. Phil. Soc., 28, 266–272.
Parrondo, J. (1996) How to cheat a bad mathematician. EEC HC&M Network on Complexity and Chaos. #ERBCHRX- CT940546. ISI, Torino, Italy. Unpublished. Available at
Paterson, A.LT. (1988) Amenability. American Mathematical Society, Providence, RI. MR: 90e:43001
Paul, A. and Pippenger, N. (2011) A census of vertices by generations in regular tessellations of the plane. Electron. J. Combin., 18(1), Research Paper 87, 13 pp. MR: 2795768
Peierls, R. (1936) On Ising's model of ferromagnetism. Math. Proc. Cambridge Philos. Soc., 32, 477–481.
Peköz, E.A and Röllin, A. (2011) New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab., 39(2), 587–608. MR: 2789507
Pemantle, R. (1988) Phase transition in reinforced random walk and RWRE on trees. Ann. Probab., 16(3), 1229–1241. MR: 89g:60220
Pemantle, R. (1991) Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., 19(4), 1559–1574. MR: 92g:60014
Pemantle, R. (2000) Towards a theory of negative dependence. J. Math. Phys., 41(3), 1371–1390. MR: 2001g:62039
Pemantle, R. and Peres, Y. (1995a) Critical random walk in random environment on trees. Ann. Probab., 23(1), 105–140. MR: 96f:60123
Pemantle, R. and Peres, Y. (1995b) Galton-Watson trees with the same mean have the same polar sets. Ann. Probab., 23(3), 1102–1124. MR: 96i:60093
Pemantle, R. and Peres, Y. (1996) On which graphs are all random walks in random environments transient? In Aldous, D. and Pemantle, R., editors, Random Discrete Structures, pages 207–211. Springer, New York. Papers from the workshop held in Minneapolis, Minnesota, November 15–19, 1993. MR: 97f:60212
Pemantle, R. and Peres, Y. (2000) Nonamenable products are not treeable. Israel J. Math., 118, 147–155. MR: 2001j:43002
Pemantle, R. and Peres, Y. (2010) The critical Ising model on trees, concave recursions and nonlinear capacity. Ann. Probab., 38(1), 184–206. MR: 2599197
Pemantle, R. and Stacey, A.M. (2001) The branching random walk and contact process on Galton-Watson and nonhomogeneous trees. Ann. Probab., 29(4), 1563–1590. MR: 1880232
Peres, Y. (1996) Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys., 177(2), 417–434. MR: 98k:60143
Peres, Y. (1999) Probability on trees: An introductory climb. In Bernard, P., editor, Lectures on Probability Theory and Statistics, vol. 1717 of Lecture Notes in Math., pages 193–280. Springer, Berlin. Lectures from the 27th Summer School on Probability Theory held in Saint-Flour, July 7–23, 1997. MR: 1746302
Peres, Y. (2000) Percolation on nonamenable products at the uniqueness threshold. Ann. Inst. H. Poincaré Probab. Statist., 36(3), 395–406. MR: 2001f:60114
Peres, Y., Pete, G., and Scolnicov, A. (2006) Critical percolation on certain nonunimodular graphs. New York J. Math., 12, 1–18 (electronic). MR: 2217160
Peres, Y., Stauffer, A., and Steif, J.E. (2015) Random walks on dynamical percolation: mixing times, mean squared displacement and hitting times. Probab. Theory Related Fields, 162(3–4), 487–530. MR: 3383336
Peres, Y. and Steif, J.E. (1998) The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields, 111(1), 141–165. MR: 99e:60217
Peres, Y. and Zeitouni, O. (2008) A central limit theorem for biased random walks on Galton-Watson trees. Probab. Theory Related Fields, 140(3–4), 595–629. MR: 2365486
Perlin, A. (2001) Probability Theory on Galton-Watson Trees. Ph.D. thesis, Massachusetts Institute of Technology. Available at
Pesin, Y.B (1997) Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics. University of Chicago Press, Chicago. MR: 1489237
Pete, G. (2008) A note on percolation on.d: Isoperimetric profile via exponential cluster repulsion. Electron. Commun. Probab., 13, 377–392. MR: 2415145
Petersen, K. (1983) Ergodic Theory. Cambridge University Press, Cambridge. MR: 87i:28002
Peyre, R. (2008) A probabilistic approach to Carne's bound. Potential Anal., 29(1), 17–36. MR: 2421492
Piau, D. (1996) Functional limit theorems for the simple random walk on a supercritical Galton-Watson tree. In Chauvin, B., Cohen, S., and Rouault, A., editors, Trees, vol. 40 of Progr. Probab., pages 95–106. Birkhäuser, Basel. Proceedings of the Workshop held in Versailles, June 14–16, 1995. MR: 1439974
Piau, D. (1998) Théoreme central limite fonctionnel pour une marche au hasard en environnement aléatoire. Ann. Probab., 26(3), 1016–1040. MR: 1634413
Piau, D. (2002) Scaling exponents of random walks in random sceneries. Stochastic Process. Appl., 100, 3–25. MR: 1919605
Pinsker, M.S (1973) On the complexity of a concentrator. In Proceedings of the Seventh International Teletraffic Congress (Stockholm, 1973), pages 318/1–318/4.
Pitman, J. (1993) Probability. Springer, New York.
Pitman, J. (1998) Enumerations of trees and forests related to branching processes and random walks. In Aldous, D. and Propp, J., editors, Microsurveys in Discrete Probability, vol. 41 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 163–180. Amer. Math. Soc., Providence, RI. Papers from the workshop held as part of the Dimacs Special Year on Discrete Probability in Princeton, NJ, June 2–6, 1997. MR: 1630413
Pitt, L.D (1982) Positively correlated normal variables are associated. Ann. Probab., 10(2), 496–499. MR: 665603
Pittet, C. and Saloff-Coste, L. (1999) Amenable groups, isoperimetric profiles and random walks. In Geometric Group Theory Down Under (Canberra, 1996), pages 293–316. de Gruyter, Berlin. MR: 1714851
Pittet, C. and Saloff-Coste, L. (2001) A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples. In preparation. Preliminary version available at
Poghosyan, V.S, Priezzhev, V.B., and Ruelle, P. (2011) Return probability for the loop-erased random walk and mean height in the Abelian sandpile model: A proof. J. Stat. Mech.: Theory Experiment, 2011(10), P10004.
Pólya, G. (1921) Ü ber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strasennetz. Math. Ann., 84(1–2), 149–160. MR: 1512028
Ponzio, S. (1998) The combinatorics of effective resistances and resistive inverses. Inform. and Comput., 147(2), 209–223. MR: 1662276
Port, S.C and Stone, C.J. (1978) Brownian Motion and Classical Potential Theory. Probability and Mathematical Statistics. Academic Press [Harcourt Brace Jovanovich Publishers], New York. MR: 0492329
Propp, J.G and Wilson, D.B. (1998) How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. J. Algorithms, 27(2), 170–217. 7th Annual ACM-SIAM Symposium on Discrete Algorithms (Atlanta, GA, 1996). MR: 99g:60116
Puder, D. (2015) Expansion of random graphs: New proofs, new results. Invent. Math., 201(3), 845–908. MR: 3385636
Pyke, R. (2003) On random walks and diffusions related to Parrondo's games. In Moore, M., Froda, S., and Léger, C., editors, Mathematical Statistics and Applications: Festschrift for Constance van Eeden, vol. 42 of IMS Lecture Notes Monogr. Ser., pages 185–216. Inst. Math. Statist., Beachwood, OH. MR: 2138293
Ratcliffe, J.G (2006) Foundations of Hyperbolic Manifolds, 2nd ed. Vol. 149 of Graduate Texts in Mathematics. Springer, New York. MR: 2249478
Raugi, A. (2004) A general Choquet-Deny theorem for nilpotent groups. Ann. Inst. H. Poincaré Probab. Statist., 40(6), 677–683. MR: 2096214
Reimer, D. (2000) Proof of the van den Berg-Kesten conjecture. Combin. Probab. Comput., 9(1), 27–32. MR: 1751301
Revelle, D. (2001) Biased random walk on lamplighter groups and graphs. J. Theoret. Probab., 14(2), 379–391. MR: 1838 734
Révész, P. (2005) Random Walk in Random and Non-Random Environments, 2nd ed. World Scientific, Hackensack, NJ. MR: 2168855
Rider, B. and Virág, B. (2007) The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN, 2007, Art. rnm006, 33. MR: 2361453
Rokhlin, V.A (1949) On the fundamental ideas of measure theory (Russian). Mat. Sbornik N.S., 25(67), 107–150. English translation: Amer. Math. Soc. Translation 1952 (1952), no. 71, 54 pp. MR: 0030584
Rosenblatt, J. (1981) Ergodic and mixing random walks on locally compact groups. Math. Ann., 257(1), 31–42. MR: 630645
Rosenblatt, M. (1971) Markov Processes. Structure and Asymptotic Behavior. Vol. 184 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York. MR: 48:7379
Ross, S.M (1996) Stochastic Processes, 2nd ed., John Wiley, New York. MR: 97a:60002
Ross, S.M and Peköz, E.A. (2007) A Second Course in Probability., Boston.
Royden, H.L (1952) Harmonic functions on open Riemann surfaces. Trans. Amer. Math. Soc., 73, 40–94. MR: 0049396
Royden, H.L (1988) Real Analysis, 3rd ed. Macmillan, New York. MR: 90g:00004
Rudin, W. (1987) Real and Complex Analysis, 3rd ed. McGraw-Hill, New York. MR: 88k:00002
Rudin, W. (1991) Functional Analysis, 2nd ed. International Series in Pure and Applied Mathematics. McGraw-Hill, New York. MR: 92k:46001
Sabidussi, G. (1964) Vertex-transitive graphs. Monatsh. Math., 68, 426–438. MR: 0175815
Sabot, C. and Tarres, P. (2015) Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. (JEMS), 17(9), 2353–2378. MR: 3420510
Saloff-Coste, L. (1995) Isoperimetric inequalities and decay of iterated kernels for almost-transitive Markov chains. Combin. Probab. Comput., 4(4), 419–442. MR: 1377559
Salvatori, M. (1992) On the norms of group-invariant transition operators on graphs. J. Theoret. Probab., 5(3), 563–576. MR: 93h:60113
Sava, E. (2010a) Lamplighter Random Walks and Entropy-Sensitivity of Languages. Ph.D. thesis, Technische Universität Graz. Available at
Sava, E. (2010b) A note on the Poisson boundary of lamplighter random walks. Monatsh. Math., 159(4), 379–396. MR: 2600904
Sawyer, S.A (1997) Martin boundaries and random walks. In Korányi, A., editor, Harmonic Functions on Trees and Buildings, vol. 206 of Contemp. Math., pages 17–44. Amer. Math. Soc., Providence, RI. Proceedings of the Workshop on Harmonic Functions on Graphs held at City University of New York, New York, October 30–November 3, 1995. MR: 1463727
Schlichting, G. (1979) Polynomidentitäten und Permutationsdarstellungen lokalkompakter Gruppen. Invent. Math., 55(2), 97–106. MR: 81d:22006
Schoenberg, I.J (1937) On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space. Ann. of Math. (2), 38(4), 787–793. MR: 1503370
Schoenberg, I.J (1938) Metric spaces and positive definite functions. Trans. Amer. Math. Soc., 44(3), 522–536. MR: 1501980
Schonmann, R.H (1992) On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab., 20(1), 174–193. MR: 1143417
Schonmann, R.H (1999a) Percolation in 1 + 1 dimensions at the uniqueness threshold. In Bramson, M. and Durrett, R., editors, Perplexing Problems in Probability, pages 53–67. Birkhäuser, Boston. Festschrift in honor of Harry Kesten. MR: 1703 124
Schonmann, R.H (1999b) Stability of infinite clusters in supercritical percolation. Probab. Theory Related Fields, 113(2), 287–300. MR: 1676831
Schonmann, R.H (2001) Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys., 219(2), 271–322. MR: 2002h:82036
Schramm, O. (2000) Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118, 221–288. MR: 1776 084
Schrijver, A. (2003) Combinatorial Optimization. Polyhedra and Efficiency. Vol. B. Vol. 24 of Algorithms and Combinatorics. Springer-Verlag, Berlin. Matroids, trees, stable sets, Chapters 39–69,/sp>. MR: 1956925
Seneta, E. (1968) On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist., 39, 2098–2102. MR: 38:2847
Seneta, E. (1970) On the supercritical Galton-Watson process with immigration. Math. Biosci., 7, 9–14. MR: 42:5348
Sheffield, S. (2007) Gaussian free fields for mathematicians. Probab. Theory Related Fields, 139(3–4), 521–541. MR: 2322706
Shepp, L.A (1972) Covering the circle with random arcs. Israel J. Math., 11, 328–345. MR: 45:4468
Shields, P.C (1987) The ergodic and entropy theorems revisited. IEEE Trans. Inform. Theory, 33(2), 263–266. MR: 880168
Shrock, R. and Wu, F.Y. (2000) Spanning trees on graphs and lattices in d dimensions. J. Phys. A, 33(21), 3881–3902. MR: 2001b:05111
Singh, M. and Vishnoi, N.K. (2014) Entropy, optimization and counting. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC '14, pages 50–59. ACM, New York.
Slade, G. (2011) The self-avoiding walk: A brief survey. In Surveys in Stochastic Processes, EMS Ser. Congr. Rep., pages 181–199. Eur. Math. Soc., Zürich. MR: 2883859
Smirnov, S. (2010) Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2), 172(2), 1435–1467. MR: 2680496
Soardi, P.M (1993) Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. Amer. Math. Soc., 119(4), 1239–1248. MR: 94a:31004
Soardi, P.M and Woess, W. (1990) Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z., 205(3), 471–486. MR: 91m:43002
Solomon, F. (1975) Random walks in a random environment. Ann. Probab., 3, 1–31. MR: 50:14943? Spakulová, I.
Solomon, F. (2009) Critical percolation of virtually free groups and other tree-like graphs. Ann. Probab., 37(6), 2262–2296. MR: 2573558
Spielman, D.A and Teng, S.H. (2007) Spectral partitioning works: Planar graphs and finite element meshes. Linear Algebra Appl., 421(2–3), 284–305. MR: 2294342
Spitzer, F. (1962) Hitting probabilities. J. Math. Mech., 11, 593–614. MR: 0139219
Spitzer, F. (1964) Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 3, 110–121. MR: 0172343
Spitzer, F. (1976) Principles of Random Walk, 2nd ed. Vol. 34 of Graduate Texts in Mathematics. Springer-Verlag, New York. MR: 52:9383
Stacey, A.M (1996) The existence of an intermediate phase for the contact process on trees. Ann. Probab., 24(4), 1711–1726. MR: 97j:60191
Starr, N. (1966) Operator limit theorems. Trans. Amer. Math. Soc., 121, 90–115. MR: 0190757
Steele, J.M (1997) Probability Theory and Combinatorial Optimization. Vol. 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. MR: 1422018
Strassen, V. (1965) The existence of probability measures with given marginals. Ann. Math. Statist., 36, 423–439. MR: 31:1693
Swart, J.M (2009) The contact process seen from a typical infected site. J. Theoret. Probab., 22(3), 711–740. MR: 2530110
Székely, G.J and Rizzo, M.L. (2005a) Hierarchical clustering via joint between-within distances: Extending Ward's minimum variance method. J. Classification, 22(2), 151–183. MR: 2231170
Székely, G.J and Rizzo, M.L. (2005b) A new test for multivariate normality. J. Multivariate Anal., 93(1), 58–80. MR: 2119764
Székely, G.J, Rizzo, M.L., and Bakirov, N.K. (2007) Measuring and testing dependence by correlation of distances. Ann. Statist., 35(6), 2769–2794. MR: 2382665
Sznitman, A.S (1998) Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics. Springer-Verlag, Berlin. MR: 1717054
Sznitman, A.S (2004) Topics in random walks in random environment. In Lawler, G.F., editor, School and Conference on Probability Theory, ICTP Lect. Notes, XVII, pages 203–266 (electronic). Abdus Salam Int. Cent. Theoret. Phys., Trieste. Expanded lecture notes from the school and conference held in Trieste, May 2002. MR: 2198849
Takacs, C. (1997) Random walk on periodic trees. Electron. J. Probab., 2, paper no. 1, 1–16 (electronic). MR: 1436761
Takacs, C. (1998) Biased random walks on directed trees. Probab. Theory Related Fields, 111(1), 123–139. MR: 1626778
Tanny, D. (1988) A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stochastic Process. Appl., 28(1), 123–139. MR: 90e:60105
Tanushev, M. and Arratia, R. (1997) A note on distributional equality in the cyclic tour property for Markov chains. Combin. Probab. Comput., 6(4), 493–496. MR: 1483432
Taylor, A.E and Lay, D.C. (1980) Introduction to Functional Analysis, 2nd ed. John Wiley, New York. MR: 564653
Taylor, S.J (1955) The dimensional measure of the graph and set of zeros of a Brownian path. Proc. Cambridge Philos. Soc., 51, 265–274. MR: 17,595b
Temperley, H.NV. (1974) Enumeration of graphs on a large periodic lattice. In McDonough, T.P. and Mavron, V.C., editors, Combinatorics (Proc. British Combinatorial Conf., Univ. Coll. Wales, Aberystwyth, 1973), vol. 13 of London Math. Soc. Lecture Note Ser., pages 155–159. Cambridge University Press, London. MR: 0347616
Tetali, P. (1991) Random walks and the effective resistance of networks. J. Theoret. Probab., 4(1), 101–109. MR: 92c:60097
Tetali, P. (1994a) Design of on-line algorithms using hitting times. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 402–411. ACM, New York. Held in Arlington, Virginia, January 23–25, 1994. MR: 1285184
Tetali, P. (1994b) An extension of Foster's network theorem. Combin. Probab. Comput., 3(3), 421–427. MR: 1300977
Thom, A. (2015) A remark about the spectral radius. Int. Math. Res. Not. IMRN, 2015(10), 2856–2864. MR: 3352259
Thom, A. (2016) The expected degree of minimal spanning forests. Combinatorica. Online
Thomassen, C. (1989) Transient random walks, harmonic functions, and electrical currents in infinite electrical networks. Technical Report Mat-Report n. 1989-07, Technical University of Denmark.
Thomassen, C. (1990) Resistances and currents in infinite electrical networks. J. Combin. Theory Ser. B, 49(1), 87–102. MR: 91d:94029
Thomassen, C. (1992) Isoperimetric inequalities and transient random walks on graphs. Ann. Probab., 20(3), 1592–1600. MR: 1175279
Thorisson, H. (2000) Coupling, Stationarity, and Regeneration. Probability and Its Applications. Springer-Verlag, New York. MR: 1741181
Timár, Á. (2006a) Ends in free minimal spanning forests. Ann. Probab., 34(3), 865–869. MR: 2243871
Timár, Á. (2006b) Neighboring clusters in Bernoulli percolation. Ann. Probab., 34(6), 2332–2343. MR: 2294984
Timár, Á. (2006c) Percolation on nonunimodular transitive graphs. Ann. Probab., 34(6), 2344–2364. MR: 2294985
Timár, Á. (2007) Cutsets in infinite graphs. Combin. Probab. Comput., 16(1), 159–166. MR: 2286517
Timár, Á. (2015) Indistinguishability of components of random spanning forests. Preprint,
Tjur, T. (1991) Block designs and electrical networks. Ann. Statist., 19(2), 1010–1027. MR: 1105858
Tongring, N. (1988) Which sets contain multiple points of Brownian motion?, Math. Proc. Cambridge Philos. Soc., 103(1), 181–187. MR: 913461
Tricot, C. (1982) Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc., 91(1), 57–74. MR: 633256
Trofimov, V.I (1984) Graphs with polynomial growth. Mat. Sb. (N.S.), 123(165)(3), 407–421. English translation: Math. USSR-Sb. 51 (1985), no. 2, 405–417. MR: 735714
Trofimov, V.I (1985) Groups of automorphisms of graphs as topological groups. Mat. Zametki, 38(3), 378–385, 476. English translation: Math. Notes 38 (1985), no. 3-4, 717–720. MR: 87d:05091
Truemper, K. (1989) On the delta-wye reduction for planar graphs. J. Graph Theory, 13(2), 141–148. MR: 90c:05078
Tutte, W.T (1961) On the problem of decomposing a graph into n connected factors. J. London Math. Soc., 36, 221–230. MR: 0140438
Uchiyama, K. (1998) Green's functions for random walks on ZN. Proc. London Math. Soc. (3), 77(1), 215–240. MR: 1625467
Varopoulos, N. (1985a) Isoperimetric inequalities and Markov chains. J. Funct. Anal., 63(2), 215–239. MR: 87e:60124
Varopoulos, N. (1985b) Long range estimates for Markov chains. Bull. Sci. Math. (2), 109(3), 225–252. MR: 87j:60100
Varopoulos, N. (1986) Théorie du potentiel sur des groupes et des variétés. C. R. Acad. Sci. Paris Sér. I Math., 302(6), 203–205. MR: 832044
Vatutin, V.A and Zubkov, A.M. (1985) Branching processes. I. In Teoriya Veroyatnostei. Matematicheskaya Statistika. Teoreticheskaya Kiber- netika. Tom 23, Itogi Nauki i Tekhniki, pages 3–67, 154. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow. English translation in J. Soviet Math. 39, no. 1, pp. 2431–2475. MR: 810404
Vatutin, V.A and Zubkov, A.M. (1993) Branching processes. II. J. Soviet Math., 67(6), 3407–3485. MR: 1260986
Vershik, A.M and Kaimanovich, V.A. (1979) Random walks on groups: Boundary, entropy, uniform distribution. Dokl. Akad. Nauk SSSR, 249(1), 15–18. MR: 553972
Virág, B. (2000a) Anchored expansion and random walk. Geom. Funct. Anal., 10(6), 1588–1605. MR: 1810 755
Virág, B. (2000b) On the speed of random walks on graphs. Ann. Probab., 28(1), 379–394. MR: 2001g:60173
Virág, B. (2002) Fast graphs for the random walker. Probab. Theory Related Fields, 124(1), 50–72. MR: 1929811
Volkov, S. (2001) Vertex-reinforced random walk on arbitrary graphs. Ann. Probab., 29(1), 66–91. MR: 1825142
Walters, P. (1982) An Introduction to Ergodic Theory. Springer-Verlag, New York. MR: 84e:28017
Watkins, M.E (1970) Connectivity of transitive graphs. J. Combinatorial Theory, 8, 23–29. MR: 0266804
Waymire, E.C and Williams, S.C. (1996) A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc., 348(2), 585–632. MR: 1322959
Whitman, W.W (1964) Some Strong Laws for Random Walks and Brownian Motion. Ph.D. thesis, Cornell University. MR: 2614450
Whitney, H. (1932) Congruent graphs and the connectivity of graphs. Amer. J. Math., 54(1), 150–168. MR: 1506881
Whyte, K. (1999) Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture. Duke Math. J., 99(1), 93–112. MR: 1700742
Wieland, B. and Wilson, D.B. (2003) Winding angle variance of Fortuin-Kasteleyn contours. Phys. Rev. E, 68, 056101.
Wilkinson, D. and Willemsen, J.F. (1983) Invasion percolation: A new form of percolation theory. J. Phys. A, 16(14), 3365–3376. MR: 725616
Wilson, D.B (1996) Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pages 296–303. ACM, New York. Held in Philadelphia, PA, May 22–24, 1996. MR: 1427525
Wilson, D.B (1997) Determinant algorithms for random planar structures. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans, LA, 1997), pages 258–267. ACM, New York. Held in New Orleans, LA, January 5–7, 1997. MR: 1447 672
Wilson, D.B (2004a) Red-green-blue model. Phys. Rev. E (3), 69, 037105.
Wilson, J.S (2004b) On exponential growth and uniformly exponential growth for groups. Invent. Math., 155(2), 287–303. MR: 2031429
Wilson, J.S (2004c) Further groups that do not have uniformly exponential growth. J. Algebra, 279(1), 292–301. MR: 2078400
Winkler, R.L (1969) Scoring rules and the evaluation of probability assessors. J. Amer. Stat. Assoc., 64(327), 1073–1078.
Woess, W. (1991) Topological groups and infinite graphs. Discrete Math., 95(1–3), 373–384. MR: 93i:22004
Woess, W. (2000) RandomWalks on Infinite Graphs and Groups. Vol. 138 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge. MR: 2001k:60006
Wolf, J.A (1968) Growth of finitely generated solvable groups and curvature of Riemanniann manifolds. J. Differential Geometry, 2, 421–446. MR: 0248688
Yaglom, A.M (1947) Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.), 56, 795–798. MR: 9,149e
Zeitouni, O. (2004) Random walks in random environment. In Lectures on Probability Theory and Statistics, vol. 1837 of Lecture Notes in Math., pages 189–312. Springer, Berlin. Lectures from the 31st Summer School on Probability Theory held in Saint-Flour, July 8–25, 2001, edited by Jean Picard. MR: 2071631
Zemanian, A.H (1976) Infinite electrical networks. Proc. IEEE, 64(1), 6–17. Recent trends in system theory. MR: 0453371
Zubkov, A.M (1975) Limit distributions of the distance to the nearest common ancestor. Teor. Verojatnost. i Primenen., 20(3), 614–623. English translation: Theor. Probab. Appl. 20, 602–612. MR: 53:1770
Zucker, I.J (2011) 70+ years of the Watson integrals. J. Stat. Phys., 145(3), 591–612. MR: 2862945
Zuk, A. (1996) La propriété (T) de Kazhdan pour les groupes agissant sur les polyedres. C. R. Acad. Sci. Paris Sér. I Math., 323(5), 453–458. MR: 1408975