Home
• Get access
• Cited by 33
• Cited by
##### This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

JUSCHENKO, KATE MATTE BON, NICOLÁS MONOD, NICOLAS and DE LA SALLE, MIKAEL 2018. Extensive amenability and an application to interval exchanges. Ergodic Theory and Dynamical Systems, Vol. 38, Issue. 01, p. 195.

Reddy, Tulasi Ram Vadlamani, Sreekar and Yogeshwaran, D. 2018. Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs. Journal of Statistical Physics,

Peres, Yuval and Zheng, Tianyi 2018. On Groups, Slow Heat Kernel Decay Yields Liouville Property and Sharp Entropy Bounds. International Mathematics Research Notices,

Camby, E. Caporossi, G. Paiva, M. H. M. and Segatto, M. E. V. 2018. Expected distance based on random walks. Journal of Mathematical Chemistry, Vol. 56, Issue. 2, p. 618.

Hermon, Jonathan 2018. Infinite and Giant Components in the Layers Percolation Model . Latin American Journal of Probability and Mathematical Statistics, Vol. 15, Issue. 1, p. 121.

Weihrauch, Tobias 2018. A Characterization of Effective Resistance Metrics. Potential Analysis,

Avena, Luca Castell, Fabienne Gaudillière, Alexandre and Mélot, Clothilde 2018. Random Forests and Networks Analysis. Journal of Statistical Physics,

Járai, Antal A. 2018. Probabilistic Cellular Automata. Vol. 27, Issue. , p. 79.

Hladký, Jan Nachmias, Asaf and Tran, Tuan 2018. The Local Limit of the Uniform Spanning Tree on Dense Graphs. Journal of Statistical Physics,

Gwynne, Ewain Kassel, Adrien Miller, Jason and Wilson, David B. 2018. Active Spanning Trees with Bending Energy on Planar Maps and SLE-Decorated Liouville Quantum Gravity for $${\kappa > 8}$$κ>8. Communications in Mathematical Physics, Vol. 358, Issue. 3, p. 1065.

Bücking, Ulrike 2018. On Rigidity and Convergence of Circle Patterns. Discrete & Computational Geometry,

Bowditch, Adam 2018. Escape regimes of biased random walks on Galton–Watson trees. Probability Theory and Related Fields, Vol. 170, Issue. 3-4, p. 685.

Miller, Joseph S. and Rute, Jason 2018. Energy randomness. Israel Journal of Mathematics,

van den Berg, Jacob and Bethuelsen, Stein Andreas 2018. Stochastic domination in space-time for the contact process. Random Structures & Algorithms, Vol. 53, Issue. 2, p. 221.

Hu, Yueyun and Shi, Zhan 2018. The Free Energy in the Derrida–Retaux Recursive Model. Journal of Statistical Physics, Vol. 172, Issue. 3, p. 718.

Külske, C. and Schriever, P. 2018. Non-robust Phase Transitions in the Generalized Clock Model on Trees. Journal of Statistical Physics, Vol. 170, Issue. 1, p. 1.

Graf, Robert 2017. Self-destructive percolation as a limit of forest-fire models on regular rooted trees. Random Structures & Algorithms, Vol. 50, Issue. 1, p. 86.

Richier, Loïc 2017. Limits of the boundary of random planar maps. Probability Theory and Related Fields,

Albin, Nathan and Poggi-Corradini, Pietro 2017. Geometric Function Theory in Higher Dimension. Vol. 26, Issue. , p. 129.

2017. Controlled Branching Processes. p. 197.

×

#### 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.

#### Reviews

‘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 (www.maa.org)

'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

#### 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.
×
Bibliography
, , and (2016) The measurable Kesten theorem. Ann. Probab., 44(3), 1601–1646.
and (2015) Fixed-energy harmonic functions. Preprint, :http://www.arxiv.org/abs/1505.05785.
and (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 http://portal.acm.org/toc.cfm?id=380752. MR: 2120364
and (2007) Fast computation of low-rank matrix approximations. J. ACM, 54(2), Art. 9. MR: 2295993
(1988) Indecomposability of treed equivalence relations. Israel J. Math., 64(3), 362–380. MR: 995576
(1990) Trees and amenable equivalence relations. Ergodic Theory Dynam. Systems, 10(1), 1–14. MR: 91d:28041
and (1991) Amenability, Kazhdan's property and percolation for trees, groups and equivalence relations. Israel J. Math., 75(2–3), 341–370. MR: 93j:43001
and (1990) Kazhdan groups, cocycles and trees. Amer. J. Math., 112(2), 271–287. MR: 91c:22011
and (2003) Bootstrap percolation: Visualizations and applications. Brazilian J. Phys., 33(3), 641–644. http://dx.doi.org/10.1590/S0103-97332003000300031.
(2008) Transient random walks in random environment on a Galton-Watson tree. Probab. Theory Related Fields, 142(3–4), 525–559. MR: 2438700
(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
(2011) Uniform measure on a Galton-Watson tree without the X log X condition. Preprint, http://www.arxiv.org/abs/1101.1816.
(2013) Note on the mononicity of the speed of the biased random walk on a Galton-Watson tree. http://www.proba.jussieu.fr/dw/lib/exe/fetch.php?media=users:aidekon:noteaidekon.pdf.
(2014) Speed of the biased random walk on a Galton-Watson tree. Probab. Theory Related Fields, 159(3–4), 597–617. MR: 3230003
(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
and (1987) Sharpness of the phase transition in percolation models. Comm. Math. Phys., 108(3), 489–526. MR: 88c:82026
, , , and (1999) Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms, 15(3–4), 319–367. MR: 2001c:60151
, , , and (1988) Discontinuity of the magnetization in one-dimensional Ising and Potts models. J. Statist. Phys., 50(1–2), 1–40. MR: 89f:82072
, , and (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
and (1988) Metastability effects in bootstrap percolation. J. Phys. A, 21(19), 3801–3813. MR: 968311
(1987) On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probab. Eng. Inform. Sc., 1, 33–46. http://dx.doi.org/10.1017/S0269964800000267.
(1990) The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math., 3(4), 450–465. MR: 91h:60013
(1991) Random walk covering of some special trees. J. Math. Anal. Appl., 157(1), 271–283. MR: 1109456
and (2002) Reversible Markov Chains and Random Walks on Graphs. Unfinished monograph, recompiled 2014 version available at http://www.stat.berkeley.edu/_aldous/RWG/book.html.
and (2007) Processes on unimodular random networks. Electron. J. Probab., 12, paper no. 54, 1454–1508 (electronic). MR: 2354165
, , , , and (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
(1995a) Percolation and minimal spanning forests in infinite graphs. Ann. Probab., 23(1), 87–104. MR: 96c:60114
(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
and (1994) Percolation of level sets for two-dimensional random fields with lattice symmetry. J. Statist. Phys., 77(3–4), 627–643. MR: 95i:82052
(1986) Eigenvalues and expanders. Combinatorica, 6(2), 83–96. MR: 88e:05077
(2003) Problems and results in extremal combinatorics. I. Discrete Math., 273(1–3), 31–53. MR: 2025940
, , , and (2007) Non-backtracking random walks mix faster. Commun. Contemp. Math., 9(4), 585–603. MR: 2348845
, , and (2004) Percolation on finite graphs and isoperimetric inequalities. Ann. Probab., 32(3A), 1727–1745. MR: 2073175
, , and (2002) The Moore bound for irregular graphs. Graphs Combin., 18(1), 53–57. MR: 1892433
and (1985) isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B, 38(1), 73–88. MR: 782626
and (2016) Speed exponents of random walks on groups. IMRN, 2016. http://dx.doi.org/10.1093/imrn/rnv378.
and (1989) A proof of the Markov chain tree theorem. Statist. Probab. Lett., 8(2), 189–192. MR: 1017890
(1988) Positive harmonic functions and hyperbolicity. In , , , and , editors, Potential Theory—Surveys and Problems (Prague, 1987), pages 1–23. Springer, Berlin. MR: 973 878
, , and (1999) Crossing estimates and convergence of Dirichlet functions along random walk and diffusion paths. Ann. Probab., 27(2), 970–989. MR: 1698991
, , , and (2016) Boundaries of planar graphs, via circle packings. Ann. Probab., 44(3), 1956–1984. MR: 3502598
and (2007) A phase transition for the metric distortion of percolation on the hypercube. Combinatorica, 27(6), 645–658. MR: 2384409
, , , and (2006) Transience of percolation clusters on wedges. Electron. J. Probab., 11, paper no. 25, 655–669 (electronic). MR: 2242658
, , and (2014) Localization for linearly edge reinforced random walks. Duke Math. J., 163(5), 889–921. MR: 3189433
, , and (2015) The non-backtracking spectrum of the universal cover of a graph. Trans. Amer. Math. Soc., 367(6), 4287–4318. MR: 3324928
, , , and (2008) Invasion percolation on regular trees. Ann. Probab., 36(2), 420–466. MR: 2393988
, , and (2013) Scaling limit of the invasion percolation cluster on a regular tree. Ann. Probab., 41(1), 229–261. MR: 3059198
and (2016) Recurrence of weak limits of excluded minor graphs. In preparation.
and (2010) Size bias, sampling, the waiting time paradox, and infinite divisibility: When is the increment indepen-dent? Preprint, http://www.arxiv.org/abs/1007.3910.
, , , , and (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
and (1983) Branching Processes. Birkhäuser, Boston. MR: 85b:60076
(1971) A note on a functional equation arising in Galton-Watson branching processes. J. Appl. Probability, 8, 589–598. MR: 45:1271
and (1972) Branching Processes. Vol. 196 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York. MR: 51:9242
(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
(1972) Entropie des groupes de type fini. C. R. Acad. Sci. Paris Sér. A-B, 275, A1363–A1366. MR: 0324741
(1974) Théorème de Choquet-Deny pour les groupes à croissance non exponentielle. C. R. Acad. Sci. Paris Sér. A, 279, 25–28. MR: 0353405
(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
(1967) Weighted sums of certain dependent random variables. Tohoku Math. J. (2), 19, 357–367. MR: 0221571
(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. http://dx.doi.org/10.1145/103418.103440.
(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
and (1999) Cut sets and normed cohomology with applications to percolation. Proc. Amer. Math. Soc., 127(2), 589–597. MR: 99g:05119
, , and (1997) Direct evaluation of thermal fluctuations in protein using a single parameter harmonic potential. Folding & Design, 2, 173–181. http://dx.doi.org/10.1016/S1359-0278(97)00024-2.
, , and (2006) A multivariate nonparametric test of independence. J. Multivariate Anal., 97(8), 1742–1756. MR: 2298886
and (2014) Electric network for non-reversible Markov chains. Preprint, http://www.arxiv.org/abs/1405.
(1992) Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal., 2(2), 137–172. MR: 1159828
and (1994) The Poisson boundary for rank one manifolds and their cocompact lattices. Forum Math., 6(3), 301–313. MR: 1269841
, , , and (2012) The sharp threshold for bootstrap percolation in all dimensions. Trans. Amer. Math. Soc., 364(5), 2667–2701. MR: 2888224
, , and (2009) Bootstrap percolation in three dimensions. Ann. Probab., 37(4), 1329–1380. MR: 2546747
, , and (2006) Bootstrap percolation on infinite trees and non-amenable groups. Combin. Probab. Comput., 15(5), 715–730. MR: 2248323
(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.
and (2010) Exponential tail bounds for loop-erased random walk in two dimensions. Ann. Probab., 38(6), 2379–2417. MR: 2683633
and (1989) Symmetric Markov chains in Zd: How fast can they move? Probab. Theory Related Fields, 82(1), 95–108. MR: 997432
(2008) Dimension and Recurrence in Hyperbolic Dynamics. Vol. 272 of Progress in Mathematics. Birkhäuser, Basel. MR: 2434246
, , and (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
(1999) Counting paths in graphs. Enseign. Math. (2), 45(1–2), 83–131. MR: 1703364
(2003) A Wilson group of non-uniformly exponential growth. C. R. Math. Acad. Sci. Paris, 336(7), 549–554. MR: 1981466
, , and (2010) On amenability of automata groups. Duke Math. J., 154(3), 575–598. MR: 2730578
and (2005) Amenability via random walks. Duke Math. J., 130(1), 39–56. MR: 2176547
(1995) Probabilistic Techniques in Analysis. Probability and Its Applications. Springer-Verlag, New York. MR: 96e:60001
and (2008) Kakeya sets in Cantor directions. Math. Res. Lett., 15(1), 73–81. MR: 2367175
and (1990) The uniformization theorem for circle packings. Indiana Univ. Math. J., 39(4), 1383–1425. MR: 1087197
and (1997) Group cohomology, harmonic functions and the first L2-Betti number. Potential Anal., 6(4), 313–326. MR: 98e:20056
, , , and (2012) Biased random walks on Galton-Watson trees with leaves. Ann. Probab., 40(1), 280–338. MR: 2917774
, , and (2014) Lyons-Pemantle-Peres monotonicity problem for high biases. Comm. Pure Appl. Math., 67(4), 519–530. MR: 3168120
, , , and (2013) Einstein relation for biased random walk on Galton-Watson trees. Ann. Inst. Henri Poincaré Probab. Stat., 49(3), 698–721. MR: 3112431
(1996) Arbres et grandes déviations. In , , and , 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
and (1992) Lectures on Hyperbolic Geometry. Universitext. Springer-Verlag, Berlin. MR: 94e:57015
(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
and (2012) Ergodic theory on stationary random graphs. Electron. J. Probab., 17, paper no. 93, 20 pp. MR: 2994841
, , , and (2015a) Disorder, entropy and harmonic functions. Ann. Probab., 43(5), 2332–2373.
, , , and (2015b) Minimal harmonic functions I, upper bounds. In preparation.
, , and (2007) Recurrence of random walk traces. Ann. Probab., 35(2), 732–738. MR: 2308594
, , , and (2004) Geometry of the uniform spanning forest: Transitions in dimensions 4; 8; 12;. Ann. of Math. (2), 160(2), 465–491. MR: 2123930
and (2005) A resistance bound via an isoperimetric inequality. Combinatorica, 25(6), 645–650. MR: 2199429
, , and (2011) A balanced excited random walk. C. R. Math. Acad. Sci. Paris, 349(7–8), 459–462. MR: 2788390
, , , and (1999a) Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab., 27(3), 1347–1356. MR: 1733 151
, , , and (1999b) Group-invariant percolation on graphs. Geom. Funct. Anal., 9(1), 29–66. MR: 99m:60149
, , , and (2001) Uniform spanning forests. Ann. Probab., 29(1), 1–65. MR: 1825 141
, , and (1999) Percolation perturbations in potential theory and random walks. In and , 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
and (2012) On the trace of branching random walks. Groups Geom. Dyn., 6(2), 231–247. MR: 2914859
, , and (2011) Is the critical percolation probability local? Probab. Theory Related Fields, 149(1–2), 261–269. MR: 2773031
, , and (1995) Martin capacity for Markov chains. Ann. Probab., 23(3), 1332–1346. MR: 96g:60098
, , and (1996) Random walks in varying dimensions. J. Theoret. Probab., 9(1), 231–244. MR: 97a:60092
, , and (1998) Unpredictable paths and percolation. Ann. Probab., 26(3), 1198–1211. MR: 99g:60183
and (1992) Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist., 28(4), 557–592. MR: 94f:60089
and (1994a) Markov chains indexed by trees. Ann. Probab., 22(1), 219–243. MR: 1258875
and (1994b) Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields, 98(1), 91–112. MR: 94m:60141
and (1996a) Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math., 126(3), 565–587. MR: 97k:31009
and (1996b) Percolation beyond Zd, many questions and a few answers. Electron. Comm. Probab., 1, paper no. 8, 71–82 (electronic). MR: 97j:60179
and (1996c) Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab., 24(3), 1219–1238. MR: 98d:60134
and (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
and (2001a) Percolation in the hyperbolic plane. J. Amer. Math. Soc., 14(2), 487–507. MR: 1815220
and (2001b) Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6, paper no. 23, 13 pp. (electronic). MR: 1873300
and (2004) Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality. In and , 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
and (2000) A random walk approach to Galton-Watson trees. J. Theoret. Probab., 13(3), 777–803. MR: 1785529
, , , and (2015) Random walks on the random graph. Preprint, http://www.arxiv.org/abs/1504.01999.
, , and (2003) The speed of biased random walk on percolation clusters. Probab. Theory Related Fields, 126(2), 221–242. MR: 1990055
and (1984) On the continuity of the percolation probability function. In , , , and , editors, Conference in Modern Analysis and Probability (New Haven, Conn., 1982), pages 61–65. Amer. Math. Soc., Providence, RI. MR: 85g:60100
and (1985) Inequalities with applications to percolation and reliability. J. Appl. Probab., 22(3), 556–569. MR: 87b:60027
and (1991) Stability properties of a flow process in graphs. Random Structures Algorithms, 2(3), 335–341. MR: 92d:90027
(1977) Chernoff's theorem in the branching random walk. J. Appl. Probability, 14(3), 630–636. MR: 0464415
, , and (1988) The spectral radius of infinite graphs. Bull. London Math. Soc., 20(2), 116–120. MR: 89a:05103
(1965) Ergodic Theory and Information. John Wiley, New York. MR: 33:254
(1995) Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. John Wiley, New York. MR: 1324786
(2014) Five remarks about random walks on groups. Preprint, http://www.arxiv.org/abs/1406.0763.
, , and (2008) Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab., 36(3), 1134–1152. MR: 2408585
(1955) On transient Markov processes with a countable number of states and stationary transition probabilities. Ann. Math. Statist., 26, 654–658. MR: 17,754d
, , and (2011) Commute times for a directed graph using an asymmetric Laplacian. Linear Algebra Appl., 435(2), 224–242. MR: 2782776
(1998) Modern Graph Theory. Vol. 184 of Graduate Texts in Mathematics. Springer-Verlag, New York. MR: 99h:05001
, , and (2009) Negative dependence and the geometry of polynomials. J. Amer. Math. Soc., 22(2), 521–567. MR: 2476782
and (1974) Strength analysis of leveling-type networks. An application of random walk theory. Geodaet. Inst. Medd., 50, 80. MR: 0475698
and (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
(1985) On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math., 52(1–2), 46–52. MR: 815600
(2004) Couplings of uniform spanning forests. Proc. Amer. Math. Soc., 132(7), 2151–2158 (electronic). MR: 2053 989
(2009) Amenability and non-uniform growth of some directed automorphism groups of a rooted tree. Math. Z., 263(2), 265–293. MR: 2534118
(2013) Behaviors of entropy on finitely generated groups. Ann. Probab., 41(6), 4116–4161. MR: 3161471
and (2015) Speed of random walks, isoperimetry and compression of finitely generated groups. Preprint, http://www.arxiv.org/abs/1510.08040.
and (1957) Percolation processes. I. Crystals and mazes. Proc. Cambridge Philos. Soc., 53, 629–641. MR: 0091567
(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. http://dx.doi.org/10.1109/SFCS.1989.63516.
and (1989) Bounds on the cover time. J. Theoret. Probab., 2(1), 101–120. MR: 981768
and (2011) Poisson boundary of GLd. Israel J. Math., 185, 125–140. MR: 2837130
, , , and (2013) Dimension (in)equalities and Hölder continuous curves in fractal percolation. J. Theoret. Probab., 26(3), 836–854. MR: 3090553
, , , and (1940) The dissection of rectangles into squares. Duke Math. J., 7, 312–340. MR: 2,153d
and (1989) Density and uniqueness in percolation. Comm. Math. Phys., 121(3), 501–505. MR: 90g:60090
and (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
(1982) A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4), 15(2), 213–230. MR: 683635
(1985) Inequalities for critical probabilities in percolation. In , 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
(1911) Cisoidal oscillations. Trans. Amer. Inst. Electrical Engineers, 30, 873–909. http://dx.doi.org/10.1109/T-AIEE.1911.4768303.
, , , and (1997) Hyperbolic geometry. In , editor, Flavors of Geometry, vol. 31 of Mathematical Sciences Research Institute Publications, pages 59–115. Cambridge University Press, Cambridge. MR: 1491098
, , and (1994) Squaring rectangles: The finite Riemann mapping theorem. In , , and , 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
, , and (2010) Proof of Aldous' spectral gap conjecture. J. Amer. Math. Soc., 23(3), 831–851. MR: 2629990
(1967) Selected Problems on Exceptional Sets. Vol. 13 of Van Nostrand Mathematical Studies. D. Van Nostrand, Princeton, NJ. MR: 0225986
, , , , , S., , , and (1968) Problems and Solutions: Advanced Problems: 5600–5609. Amer. Math. Monthly, 75(6), 685–687. MR: 1534960
(2012) A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy. Potential Anal., 37(3), 229–245. MR: 2969301
and (2015) Every planar graph with the Liouville property is amenable. Preprint, http://www.arxiv.org/abs/1502.02542.
(1985) A transmutation formula for Markov chains. Bull. Sci. Math. (2), 109(4), 399–405. MR: 87m:60142
, , and (1994) Random walks on the affine group of local fields and of homogeneous trees. Ann. Inst. Fourier (Grenoble), 44(4), 1243–1288. MR: 1306556
and (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
(1889) A theorem on trees. Quart. J. Math., 23, 376–378. http://dx.doi.org/10.1017/CBO9780511703799.010.
(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
and (1996) Planar Cayley graphs with regular dual. Internat. J. Algebra Comput., 6(5), 553–561. MR: 98a:05077
, , and (1979) Bootstrap percolation on a Bethe lattice. J. Phys. C, 12(1), L31–L35. http://dx.doi.org/10.1088/ 0022-3719/12/1/008.
, , , , and (1996/1997) The electrical resistance of a graph captures its commute and cover times. Comput. Complexity, 6(4), 312–340. MR: 99h:60140
, , and (1991) Growing conditioned trees. Stochastic Process. Appl., 39(1), 117–130. MR: 1135089
, , and (1988) Connectivity properties of Mandelbrot's percolation process. Probab. Theory Related Fields, 77(3), 307–324. MR: 931500
, , and (1985) The stochastic geometry of invasion percolation. Comm. Math. Phys., 101(3), 383–407. MR: 87i:82072
(1995a) Aspects of the fractal percolation process. In , , and , 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
(1995b) On the absence of directed fractal percolation. J. Phys. A.: Math. Gen., 28, L295–L301. http://dx.doi.org/10.1088/0305-4470/28/10/003.
, , and (1997) No directed fractal percolation in zero area. J. Statist. Phys., 88(5–6), 1353–1362. MR: 1478072
(1970) A lower bound for the smallest eigenvalue of the Laplacian. In , 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
and (1986) L2-cohomology and group cohomology. Topology, 25(2), 189–215. MR: 87i:58161
(1997) Average properties of random walks on Galton-Watson trees. Ann. Inst. H. Poincaré Probab. Statist., 33(3), 359–369. MR: 1457056
and (2004) Anchored expansion, percolation and speed. Ann. Probab., 32(4), 2978–2995. With an appendix by Gábor Pete. MR: 2094436
and (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
and (1960) Sur l'équation de convolution. C. R. Acad. Sci. Paris, 250, 799–801. MR: 22:9808
K., , , and (1986) Some intersection theorems for ordered sets and graphs. J. Combin. Theory Ser. A, 43(1), 23–37. MR: 859293
and (1998) Isoperimetric inequalities for Cartesian products of graphs. Combin. Probab. Comput., 7(2), 141–148. MR: 2000c:05085
, , and (1995) A new approach to solving three combinatorial enumeration problems on planar graphs. Discrete Appl. Math., 60(1–3), 119–129. MR: 96e:05154
(2006) On the transience of processes defined on Galton-Watson trees. Ann. Probab., 34(3), 870–878. MR: 2243872
(2003) Graphs, networks, and linear unbiased estimates. Discrete Appl. Math., 130(3), 381–393. MR: 1999697
, , , and (1993) Random walks on weighted graphs and applications to on-line algorithms. J. Assoc. Comput. Mach., 40(3), 421–453. MR: 1370357
, , and (1996) Random walks on regular and irregular graphs. SIAM J. Discrete Math., 9(2), 301–308. MR: 1386885
, , and (1993) Collisions among random walks on a graph. SIAM J. Discrete Math., 6(3), 363–374. MR: 1229691
(1996) Ultracontractivity and Nash type inequalities. J. Funct. Anal., 141(2), 510–539. MR: 1418518
, , and (2001) A geometric approach to on-diagonal heat kernel lower bounds on groups. Ann. Inst. Fourier (Grenoble), 51(6), 1763–1827. MR: 1871289
and (1993) Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana, 9(2), 293–314. MR: 94g:58263
(1847) De l'origine et des limites de la correspondance entre l'algèbre et la géométrie. Hachette, Paris. Available at http://gallica.bnf.fr/ark:/12148/bpt6k6563896n.
and (1983) Oriented percolation in dimensions: Bounds and asymptotic formulas. Math. Proc. Cambridge Philos. Soc., 93(1), 151–162. MR: 84e:60150
(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
and (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
and (2011) Random recursive triangulations of the disk via fragmentation theory. Ann. Probab., 39(6), 2224–2270. MR: 2932668
and (2016) The harmonic measure of balls in random trees. Ann. Probab. To appear, http://www.arxiv.org/abs/1304.7190.
and (2011) Random laminations and multitype branching processes. Electron. Commun. Probab., 16, 435–446. MR: 2831082
(2005) A once edge-reinforced random walk on a Galton-Watson tree is transient. Statist. Probab. Lett., 73(2), 115–124. MR: 2159246
, , and (1963) On Weyl's criterion for uniform distribution. Michigan Math. J., 10, 311–314. MR: 0153656
and (1990) On the structure of Mandelbrot's percolation process and other random Cantor sets. J. Statist. Phys., 58(5–6), 1109–1126. MR: 1049059
and (1978) Probabilities and Potential. Vol. 29 of North-Holland Mathematics Studies. North-Holland, Amsterdam. MR: 521810
(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
, , , and (2002) Large deviations for random walks on Galton-Watson trees: Averaging and uncertainty. Probab. Theory Related Fields, 122(2), 241–288. MR: 1894069
, , and (2004) Large deviations for random walk in random environment with holding times. Ann. Probab., 32(1B), 996–1029. MR: 2044672
, , , and (2001) Thick points for planar Brownian motion and the Erdʺos-Taylor conjecture on random walk. Acta Math., 186(2), 239–270. MR: 1846031
and (1998) Large Deviations Techniques and Applications, 2nd ed. Vol. 38 of Applications of Mathematics. Springer-Verlag, New York. MR: 1619036
(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
(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
and (1997) Geometry of Cuts and Metrics. Vol. 15 of Algorithms and Combinatorics. Springer-Verlag, Berlin. MR: 1460488
and (2002) A different construction of Gaussian fields from Markov chains: Dirichlet covariances. Ann. Inst. H. Poincaré Probab. Statist., 38(6), 863–878. MR: 1955341
and (1990) Strong stationary times via a new form of duality. Ann. Probab., 18(4), 1483–1522. MR: 1071805
and (1994) Moderate growth and random walk on finite groups. Geom. Funct. Anal., 4(1), 1–36. MR: 1254308
and (2001) A conjecture concerning a limit of non-Cayley graphs. J. Algebraic Combin., 14(1), 17–25. MR: 2002h:05082
, , and (2012) Cover times, blanket times, and majorizing measures. Ann. of Math. (2), 175(3), 1409–1471. MR: 2912708
, , and (2013) Markov type and threshold embeddings. Geom. Funct. Anal., 23(4), 1207–1229. MR: 3077911
, , and (2015) Transience of edge-reinforced random walk. Comm. Math. Phys., 339(1), 121–148. MR: 3366053
(1977) de Rham-Hodge theory for L2-cohomology of infinite coverings. Topology, 16(2), 157–165. MR: 0445560
(1979) L2 harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Amer. Math. Soc., 77(3), 395–400. MR: 81e:58004
(1984) Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc., 284(2), 787–794. MR: 85m:58185
and (1986) Combinatorial Laplacians and isoperimetric inequality. In , 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
(1984) Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, New York. MR: 85k:31001
(1988) Electric currents in infinite networks. Unpublished, http://www.arxiv.org/abs/math/0703899.
(2009) The Kemeny constant of a Markov chain. Unpublished, http://www.arxiv.org/abs/0909.2636.
and (1984) Random Walks and Electric Networks. Mathematical Association of America, Washington, DC. Also available at http://arxiv.org/abs/math/0001057. MR: 89a:94023
and (2008) Commuting time geometry of ergodic Markov chains. Unpublished, http://www.arxiv.org/abs/1107.2612.
, , and (2014) An Introduction to Random Interlacements. Springer Briefs in Mathematics. Springer, Cham. MR: 3308116
and (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
(1962) The extremal length of a network. J. Math. Anal. Appl., 5, 200–215. MR: 0143468
, , and (2016) Absence of infinite cluster for critical Bernoulli percolation on slabs. Comm. Pure Appl. Math., 69(7), 1397–1411. MR: 3503025
and (2012) The connective constant of the honeycomb lattice equals. Ann. of Math. (2), 175(3), 1653–1665. MR: 2912714
and (2015) A new proof of the sharpness of the phase transition for Bernoulli percolation on.d. Enseign. Math. To appear, http://www.arxiv.org/abs/1502.03051.
and (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
(1992) Loop-erased self-avoiding walks in 2D. Physica A, 191, 516–522. http://dx.doi.org/10.1016/ 0378-4371(92)90575-B.
and (2002) Random trees, Lévy processes and spatial branching processes. Astérisque, 281, vi+147. MR: 1954248
(1986) Reversible diffusion processes. In and , 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
(2010) Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. MR: 2722836
, , and (1991) Making money from fair games. In and , 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
(1949) On the strong stability of a sequence of events. Ann. Math. Statistics, 20, 296–299. MR: 0031675
and (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
, , and (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
, , , and (1957) Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc., 53, 856–862. MR: 0094855
(1969) The total progeny in a branching process and a related random walk. J. Appl. Probability, 6, 682–686. MR: 0253433
(2010) Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups. Duke Math. J., 154(3), 509–526. MR: 2730576
(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; http://dx.doi.org/10.1070/RM1969v024n02ABEH001341. MR: 0245096
(1980) Markov processes and random fields. Bull. Amer. Math. Soc. (N.S.), 3(3), 975–999. MR: 585179
and (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
(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
(1990) Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. Springer-Verlag, New York. MR: 1065392
(1971) Matroids and the greedy algorithm. Math. Programming, 1, 127–136. MR: 0297357
(1969) On the nonexistence of uniform homeomorphisms between Lp-spaces. Ark. Mat., 8, 103–105 (1969). MR: 0271719
and (2009) Rigidity and equivalence relations with infinitely many ends. Unpublished manuscript available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.156.8077.
(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
(2011) Poisson-Furstenberg boundary of random walks on wreath products and free metabelian groups. Comment. Math. Helv., 86(1), 113–143. MR: 2745278
and (2010) Homomorphisms to. constructed from random walks. Ann. Inst. Fourier (Grenoble), 60(6), 2095–2113. MR: 2791651
, , and (2012) Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs. Ann. of Math. (2), 176(1), 221–260. MR: 2925383
and (2009) Limit theorems for Parrondo's paradox. Electron. J. Probab., 14, paper no. 62, 1827–1862. MR: 2540850
(1987) Multiple points in the sample paths of a Lévy process. Probab. Theory Related Fields, 76(3), 359–367. MR: 912660
(1986) Random fractals. Math. Proc. Cambridge Philos. Soc., 100(3), 559–582. MR: 88e:28005
(1987) Cut-set sums and tree processes. Proc. Amer. Math. Soc., 101(2), 337–346. MR: 88m:90052
(1990) Fractal Geometry. Mathematical foundations and applications. John Wiley, Chichester. MR: 1102677
(1989) Décompositions de Mesures et Recouvrements Aléatoires. Ph.D. thesis, Université de Paris-Sud, Département de Mathématique, Orsay. MR: 91e:60009
(1990) Sur quelques processus de naissance et de mort. C. R. Acad. Sci. Paris Sér. I Math., 310(6), 441–444. MR: 91d:60103
(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
, , and (2012) Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Related Fields, 154(3–4), 621–660. MR: 3000557
and (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. http://dx.doi.org/10.1145/129712.129716.
(1971) An Introduction to Probability Theory and Its Applications. Vol. II., 2nd ed. John Wiley, New York. MR: 42:5292
(1961) Critical probabilities for cluster size and percolation problems. J. Math. Phys., 2, 620–627. MR: 0126306
and (2015) Nearest-neighbor percolation function is continuous for d > 10. Preprint, http://www.arxiv.org/abs/1506.07977.
and (1989) Capacity and energy for multiparameter Markov processes. Ann. Inst. H. Poincaré Probab. Statist., 25(3), 325–350. MR: 1023955
and (2009) Analytic Combinatorics. Cambridge University Press, Cambridge. MR: 2483235
(1971) Infinite networks. I: Resistive networks. IEEE Trans. Circuit Theory, CT–18, 326–331. MR: 0275998
(1974) A new proof of R. Foster's averaging formula in networks. Linear Algebra and Appl., 8, 35–37. MR: 0329772
and (1977) The structure of reduced critical Galton-Watson processes. Math. Nachr., 79, 233–241. MR: 0461689
(1971) On the “zero-two” law. Israel J. Math., 10, 275–280. MR: 0298759
(1976) More on the “zero-two” law. Proc. Amer. Math. Soc., 61(2), 262–264 (1977). MR: 0428076
(1955) On groups with full Banach mean value. Math. Scand., 3, 243–254. MR: 18,51f
, , and (2002) Stretched exponential fixation in stochastic Ising models at zero temperature. Comm. Math. Phys., 228(3), 495–518. MR: 1918786
Jr. and (1962) Flows in Networks. Princeton University Press, Princeton, NJ. MR: 28:2917
(1972a) On the random-cluster model. II. The percolation model. Physica, 58, 393–418. MR: 51:14826
(1972b) On the random-cluster model. III. The simple random-cluster model. Physica, 59, 545–570. MR: 55:5127
and (1972) On the random-cluster model. I. Introduction and relation to other models. Physica, 57, 536–564. MR: 0359655
, , and (1971) Correlation inequalities on some partially ordered sets. Comm. Math. Phys., 22, 89–103. MR: 46:8607
(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
(1935) Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddel. Lunds Univ. Mat. Sem., 3, 1–118.
(1980) Dirichlet Forms and Markov Processes. North-Holland, Amsterdam. MR: 81f:60105
(1985) Energy forms and diffusion processes. In , editor, Mathematics + Physics. Vol. 1, pages 65–97. World Scientific, Singapore. Lectures on recent results. MR: 87m:60176
(1963) A Poisson formula for semi-simple Lie groups. Ann. of Math. (2), 77, 335–386. MR: 0146298
(1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1, 1–49. MR: 35:4369
(1970) Intersections of Cantor sets and transversality of semigroups. In , 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
(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
(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
(1973) Boundary theory and stochastic processes on homogeneous spaces. In , 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
(2008) Ergodic fractal measures and dimension conservation. Ergodic Theory Dynam. Systems, 28(2), 405–422. MR: 2408385
and (2003) Markov processes and Ramsey theory for trees. Combin. Probab. Comput., 12(5–6), 547–563. MR: 2037069
(1998) Mercuriale de groupes et de relations. C. R. Acad. Sci. Paris Sér. I Math., 326(2), 219–222. MR: 99h:28034
(2000) Coût des relations d'équivalence et des groupes. Invent. Math., 139(1), 41–98. MR: 1728 876
(2002) Invariants l2 de relations d'équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci., 95, 93–150. MR: 1953 191
(2005) Invariant percolation and harmonic Dirichlet functions. Geom. Funct. Anal., 15(5), 1004–1051. MR: 2221157
and (2009) A measurable-group-theoretic solution to von Neumann's problem. Invent. Math., 177(3), 533–540. MR: 2534099
and (2015) Approximations of standard equivalence relations and Bernoulli percolation at pu. Preprint, http://www.arxiv.org/abs/1509.00247.
, , and (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
, , , and (2012) Random walks on Galton-Watson trees with random conductances. Stochastic Process. Appl., 122(4), 1652–1671. MR: 2914767
, , and (2013) The scaling limits of the minimal spanning tree and invasion percolation in the plane. Preprint, http://www.arxiv.org/abs/1309.0269.
and (2005) Harmonic Measure. Vol. 2 of New Mathematical Monographs. Cambridge University Press, Cambridge. MR: 2150803
and (2014) A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Related Fields, 158(1–2), 55–89. MR: 3152780
and (2012) Poisson boundary of groups acting on.-trees. Israel J. Math., 191(2), 585–646. MR: 3011489
(1995) Contour processes of random trees. In , 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
(1999) Elementary new proofs of classical limit theorems for Galton-Watson processes. J. Appl. Probab., 36(2), 301–309. MR: 1724 856
(2010) Lamplighter graphs do not admit harmonic functions of finite energy. Proc. Amer. Math. Soc., 138(9), 3057–3061. MR: 2653930
(2016) The boundary of a square tiling of a graph coincides with the Poisson boundary. Invent. Math., 203(3), 773–821. MR: 3461366
and (2014) New bounds for edge-cover by random walk. Combin. Probab. Comput., 23(4), 571–584. MR: 3217361
(1988) Random walks on graphs with a strong isoperimetric property. J. Theoret. Probab., 1(2), 171–187. MR: 89g:60216
(2007) Rate of escape of random walks on free products. J. Aust. Math. Soc., 83(1), 31–54. MR: 2378433
(2011) Asymptotic entropy of random walks on free products. Electron. J. Probab., 16, paper no. 3, 76–105. MR: 2749773
and (1997) Kazhdan's property T and the geometry of the collection of invariant measures. Geom. Funct. Anal., 7(5), 917–935. MR: 99f:28029
and (1977) Extended Watson integrals for the cubic lattices. Proc. Nat. Acad. Sci. U.S.A., 74(5), 1800–1801. MR: 0442300
and (2007) Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc., 102(477), 359–378. MR: 2345548
, , , , and (1989) A note on bounded automorphisms of infinite graphs. Graphs Combin., 5(4), 333–338. MR: 1032384
and (1993) Knots, tangles, and electrical networks. Adv. in Appl. Math., 14(3), 267–306. MR: 94m:57013
, , and (2015) Sharp lower bounds for the asymptotic entropy of symmetric random walks. Groups Geom. Dyn., 9(3), 711–735. MR: 3420541
(2014) On Turing dynamical systems and the Atiyah problem. Invent. Math., 198(1), 27–69. MR: 3260857
(1987) Statistically self-similar fractals. Probab. Theory Related Fields, 74(3), 357–392. MR: 88c:60038
, , and (1988) The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc., 71(381), x+121. MR: 88k:28010
and (2009) Local bootstrap percolation. Electron. J. Probab., 14, paper no. 14, 385–399. MR: 2480546
(1980) A new look at convergence of branching processes. Ann. Probab., 8(2), 377–380. MR: 81e:60091
and (1982) Critical phenomena for Spitzer's reversible nearest particle systems. Ann. Probab., 10(4), 881–895. MR: 84f:60140
(1980) Symmetrical random walks on discrete groups. In , , and , editors, Multicomponent Random Systems, vol. 6 of Adv. Probab. Related Topics, pages 285–325. Dekker, New York. Translated from the Russian. MR: 599539
(1983) On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR, 271(1), 30–33. MR: 85g:20042
(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
(1999) Percolation, 2nd ed. Springer-Verlag, Berlin. MR: 1707 339
(2006) The Random-Cluster Model. Vol. 333 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. MR: 2243761
and (1983) Random electrical networks on complete graphs II: Proofs. Unpublished manuscript, available at http://www.arxiv.org/abs/math.PR/0107068.
, , and (1993) Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields, 96(1), 33–44. MR: 94i:60078
and (1990) The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A, 430(1879), 439–457. MR: 91m:60186
and (1990) Percolation in 1 + 1 dimensions. In and , 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
and (1998) Critical probabilities for site and bond percolation models. Ann. Probab., 26(4), 1788–1812. MR: 1675079
and (2001) Probability and Random Processes, 3rd ed. Oxford University Press, New York. MR: 2059709
(1981a) Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math., 53, 53–73. MR: 83b:53041
(1981b) Structures métriques pour les variétés riemanniennes. Vol. 1 of Textes Mathématiques [Mathematical Texts]. CEDIC, Paris. Edited by and . MR: 682063
(1983) Filling Riemannian manifolds. J. Differential Geom., 18(1), 1–147. MR: 697984
(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
(2005) Gillis's random walks on graphs. J. Appl. Probab., 42(1), 295–301. MR: 2144913
(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
and (1990) Critical exponent for the loop erased self-avoiding walk by Monte Carlo methods. J. Stat. Phys., 59(1–2), 1–9. http://dx.doi.org/10.1007/BF01015560.
(1991) Statistical Inference for Branching Processes. Wiley Series in Probability and Mathematical Statistics. John Wiley, New York. MR: 1254434
(1994) Aspects of Spatial Random Processes. Ph.D. thesis, Göteborg, Sweden.
(1995) Random-cluster measures and uniform spanning trees. Stochastic Process. Appl., 59(2), 267–275. MR: 97b:60170
(1997) Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab., 25(3), 1423–1436. MR: 98f:60207
(1998) Uniform and minimal essential spanning forests on trees. Random Structures Algorithms, 12(1), 27–50. MR: 99i:05186
(2013) Two badly behaved percolation processes on a nonunimodular graph. J. Theoret. Probab., 26(4), 1165–1180. MR: 3119989
, , and (2002) Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab., 30(1), 443–473. MR: 2003e:60220
and (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
, , and (1999) Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In and , editors, Perplexing Problems in Probability, pages 69–90. Birkhäuser, Boston. Festschrift in honor of Harry Kesten. MR: 2000b:60003
, , and (2000) The Ising model on diluted graphs and strong amenability. Ann. Probab., 28(3), 1111–1137. MR: 2001i:60169
(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
(1961a) Comparison of atom and bond percolation processes. J. Math. Phys., 2, 728–733. MR: 0130722
(1961b) The number of polygons on a lattice. Proc. Cambridge Philos. Soc., 57, 516–523. MR: 23:A814
(1978) Nonnegative entropy measures of multivariate symmetric correlations. Information and Control, 36(2), 133–156. MR: 0464499
and (1990) Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys., 128(2), 333–391. MR: 91a:82037
and (1992) The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys., 4(2), 235–327. MR: 93j:82033
and (1994) Mean-field behaviour and the lace expansion. In , 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
and (1999) Parrondo's paradox. Statist. Sci., 14(2), 206–213. MR: 1722065
and (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
(1952) First passage and recurrence distributions. Trans. Amer. Math. Soc., 73, 471–486. MR: 0052057
(1960) A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc., 56, 13–20. MR: 22:6023
(1970/71) Some dimension theorems for the sample functions of stable processes. Indiana Univ. Math. J., 20, 733–738. MR: 45:1251
(1981) Trees generated by a simple branching process. J. London Math. Soc. (2), 24(2), 373–384. MR: 83b:60072
and (1995) Hyperbolic and parabolic packings. Discrete Comput. Geom., 14(2), 123–149. MR: 1331923
(1966) Corrections and comments on the paper “A branching process allowing immigration”. J. Roy. Statist. Soc. Ser. B, 28, 213–217. MR: 33:1896b
, , and (1967) A refinement of two theorems in the theory of branching processes. Teor. Verojatnost. i Primenen., 12, 341–346. MR: 36:978
and (1993) Gaussian estimates for Markov chains and random walks on groups. Ann. Probab., 21(2), 673–709. MR: 1217561
(1970) Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Statist., 41, 739–742. MR: 40:8136
and (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
and (2003) Isoperimetric constants of regular planar graphs. Interdiscip. Inform. Sci., 9(2), 221–228. MR: 2038013
(1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 58, 13–30. MR: 0144363
, , and (2006) A stable marriage of Poisson and Lebesgue. Ann. Probab., 34(4), 1241–1272. MR: 2257646
and (1997) p-Harmonic functions on graphs and manifolds. Manuscripta Math., 94(1), 95–110. MR: 99c:31017
(2003) Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields, 125(2), 195–224. MR: 1961342
, , , , , and (2008) Chip-firing and rotor-routing on directed graphs. In and , 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
, , and (2006) Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4), 439–561 (electronic). MR: 2247919
and (2013) Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge. MR: 2978290
(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
(2000) Zero-temperature Ising spin dynamics on the homogeneous tree of degree three. J. Appl. Probab., 37(3), 736–747. MR: 1782449
(2015a) Wired cycle-breaking dynamics for uniform spanning forests. Ann. Probab. To appear, http://www.arxiv.org/abs/1504.03928.
(2015b) Interlacements and the wired uniform spanning forest. Preprint, http://www.arxiv.org/abs/1512.08509.
(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
and (2015) Indistinguishability of trees in uniform spanning forests. Probab. Theory Related Fields. To appear.
and (2016) Uniform spanning forests of planar graphs. Preprint, http://www.arxiv.org/abs/1603.07320.
and (2015) Boundaries of planar graphs: A unified approach. Preprint, http://www.arxiv.org/abs/1508.03923.
(1978) Some global properties of symmetric diffusion processes. Publ. Res. Inst. Math. Sci., 14(2), 441–486. MR: 80d:60099
(1975) On Whitney's theorem on the unique embeddability of 3-connected planar graphs. In , editor, Recent Advances in Graph Theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pages 303–306. (Loose errata). Academia, Prague. MR: 52:5462
and (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
and (2010a) Theory of minimum spanning trees. I. Mean-field theory and strongly disordered spin-glass model. Phys. Rev. E, 81(2), 021130. http://dx.doi.org/10.1103/PhysRevE.81.021130.
and (2010b) Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold. Phys. Rev. E, 81(2), 021131. http://dx.doi.org/10.1103/PhysRevE.81.021131.
and (1973) Local asymptotic laws for Brownian motion. Ann. Probab., 1, 527–549. MR: 0365732
and (1996) Cutpoints and exchangeable events for random walks. Teor. Veroyatnost. i Primenen., 41(4), 854–868. MR: 1687097
(1997) Gaussian Hilbert Spaces. Vol. 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge. MR: 1474726
and (2008) Infinite volume limit of the abelian sandpile model in dimensions d3. Probab. Theory Related Fields, 141(1–2), 181–212. MR: 2372969
and (2014) Minimal configurations and sandpile measures. J. Theoret. Probab., 27(1), 153–167. MR: 3174221
and (1989) Approximating the permanent. SIAM J. Comput., 18(6), 1149–1178. MR: 1025467
, , and (1983) Association of normal random variables and Slepian's inequality. Ann. Probab., 11(2), 451–455. MR: 690142
and (1983) Negative association of random variables, with applications. Ann. Statist., 11(1), 286–295. MR: 85d:62058
and (1982) Exact distributions of kin numbers in a Galton-Watson process. J. Appl. Probab., 19(4), 767–775. MR: 84a:60104
and (1984) Extensions of Lipschitz mappings into a Hilbert space. In , , , and , 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
and (2014) Lp-distortion and p-spectral gap of finite graphs. Bull. Lond. Math. Soc., 46(2), 329–341. MR: 3194751
and (2008) Operator Theory of Electrical Resistance Networks. Universitext. Springer-Verlag. To appear, http://www.arxiv.org/abs/0806.3881.
(1981) Une théorie combinatoire des séries formelles. Adv. in Math., 42(1), 1–82. MR: 633783
(2003) Inequality of two critical probabilities for percolation. Electron. Comm. Probab., 8, 184–187 (electronic). MR: 2042 758
, , , and (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
, , , and (1989) On the cover time of random walks on graphs. J. Theoret. Probab., 2(1), 121–128. MR: 981769
(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
(1990) Boundary and entropy of random walks in random environment. In , , and , 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
(1992) Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators. Potential Anal., 1(1), 61–82. MR: 94i:31012
(1994) The Poisson boundary of hyperbolic groups. C. R. Acad. Sci. Paris Sér. I Math., 318(1), 59–64. MR: 1260536
(1996) Boundaries of invariant Markov operators: The identification problem. In and , 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
(2000) The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2), 152(3), 659–692. MR: 1815698
(2001) Poisson boundary of discrete groups. Preprint, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.6.6675.
(2005) “Münchhausen trick” and amenability of self-similar groups. Internat. J. Algebra Comput., 15(5–6), 907–937. MR: 2197814
and (1996) The Poisson boundary of the mapping class group. Invent. Math., 125(2), 221–264. MR: 1395719
and (1998) The Poisson boundary of Teichmüller space. J. Funct. Anal., 156(2), 301–332. MR: 1636940
and (1983) Random walks on discrete groups: Boundary and entropy. Ann. Probab., 11(3), 457–490. MR: 85d:60024
and (2002) Boundary and entropy of space homogeneous Markov chains. Ann. Probab., 30(1), 323–363. MR: 1894110
(1944) Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo, 20, 706–714. MR: 7,315b
(1982) A simple proof of the ergodic theorem using nonstandard analysis. Israel J. Math., 42(4), 284–290. MR: 682311
(1985) Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan, 37(3), 391–413. MR: 87d:53082
(1986) Rough isometries and the parabolicity of Riemannian manifolds. J. Math. Soc. Japan, 38(2), 227–238. MR: 87e:53066
and (1979) Fundamentals of the Theory of Groups. Springer-Verlag, New York. Translated from the second Russian edition by Robert G. Burns. MR: 80k:20002
(2003) Boundaries and random walks on finitely generated infinite groups. Ark. Mat., 41(2), 295–306. MR: 2011923
and (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
and (2007) The Poisson boundary of lamplighter random walks on trees. Geom. Dedicata, 124, 95–107. MR: 2318539
and (2016) The looping rate and sandpile density of planar graphs. Amer. Math. Monthly, 123(1), 19–39. MR: 3453533
(1961) The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice. Physica, 27, 1209–1225. http://dx.doi.org/10.1007/978-0-8176-4842-820.
and (1982) A simple proof of some ergodic theorems. Israel J. Math., 42(4), 291–296. MR: 682312
and (2008) Random walks on periodic graphs. Trans. Amer. Math. Soc., 360(11), 6065–6087. MR: 2425703
(1995) The essence of the law of large numbers. In , 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
(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
and (1960) Finite Markov Chains. The University Series in Undergraduate Mathematics. , Princeton, NJ. MR: 0115196
(1951) Some problems in the theory of queues. J. Roy. Statist. Soc. Ser. B., 13, 151–173; discussion: 173–185. MR: 0047944
(1899) Equivalence of triangles and stars in conducting networks. Electrical World and Engineer, 34, 413–414.
(1997) Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist., 33(5), 591–618. MR: 1473567
(1998) Tilings and discrete Dirichlet problems. Israel J. Math., 105, 61–84. MR: 99m:52026
(2000a) The asymptotic determinant of the discrete Laplacian. Acta Math., 185(2), 239–286. MR: 2002g:82019
(2000b) Long-range properties of spanning trees. J. Math. Phys., 41(3), 1338–1363. MR: 1757 962
(2001) Dominos and the Gaussian free field. Ann. Probab., 29(3), 1128–1137. MR: 1872739
(2008) Height fluctuations in the honeycomb dimer model. Comm. Math. Phys., 281(3), 675–709. MR: 2415464
, , and (2000) Trees and matchings. Electron. J. Combin., 7(1), Research Paper 25, 34 pp. (electronic). MR: 2001a:05123
and (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
(1959a) Full Banach mean values on countable groups. Math. Scand., 7, 146–156. MR: 22:2911
(1959b) Symmetric random walks on groups. Trans. Amer. Math. Soc., 92, 336–354. MR: 22:253
(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
(1980) The critical probability of bond percolation on the square lattice equals 12. Comm. Math. Phys., 74(1), 41–59. MR: 82c:60179
(1982) Percolation Theory for Mathematicians. Birkhäuser, Boston. MR: 84i:60145
(1986) Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist., 22(4), 425–487. MR: 88b:60232
, , and (1966) The Galton-Watson process with mean one and finite variance. Teor. Verojatnost. i Primenen., 11, 579–611. MR: 34:6868
and (1966) A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist., 37, 1211–1223. MR: 33:6707
, , and (2000) Limsup random fractals. Electron. J. Probab., 5, paper no. 5, 24 pp. MR: 1743726
(1986) Ergodic Theory of Random Transformations. Birkhäuser, Boston. MR: 89c:58069
(1968) The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B, 30, 499–510. MR: 0254907
(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. http://dx.doi.org/10.1002/andp.18471481202.
(2010) A new proof of Gromov's theorem on groups of polynomial growth. J. Amer. Math. Soc., 23(3), 815–829. MR: 2629989
(1938) On the solution of a problem in biology. Izv. NII Matem. Mekh. Tomskogo Univ., 2, 7–12.
and (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. http://dx.doi.org/10.1007/978-94-017-2973-411.
and (1997) Global existence theorems for harmonic maps to non-locally compact spaces. Comm. Anal. Geom., 5(2), 333–387. MR: 1483983
and (2000) Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo, 7(1), 7–25. MR: 1749978
(2008) Critical percolation of free product of groups. Internat. J. Algebra Comput., 18(4), 683–704. MR: 2428151
, , and (2013) Determinants, their applications to Markov processes, and a random walk proof of Kirchhoff's matrix tree theorem. Preprint, http://www.arxiv.org/abs/1306.2059.
(2011) Percolation on a product of two trees. Ann. Probab., 39(5), 1864–1895. MR: 2884876
and (2004) An asymptotic expansion for the discrete harmonic potential. Electron. J. Probab., 9, paper no. 1, 1–17 (electronic). MR: 2041826
(2007) Connected allocation to Poisson points in.2. Electron. Comm. Probab., 12, 140–145 (electronic). MR: 2318161
and (1974) Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley-Interscience, New York. MR: 0419394
(1966) Topology. Vol. I, revised and augmented ed. Academic Press, New York. Translated from the French by J. Jaworowski. MR: 36:840
(1998) Percolation on Fuchsian groups. Ann. Inst. H. Poincaré Probab. Statist., 34(2), 151–177. MR: 1614583
(1967) The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 7, 271–288. MR: 0217893
(1980) A self-avoiding random walk. Duke Math. J., 47(3), 655–693. MR: 81j:60081
(1983) A connective constant for loop-erased self-avoiding random walk. J. Appl. Probab., 20(2), 264–276. MR: 84g:60113
(1986) Gaussian behavior of loop-erased self-avoiding random walk in four dimensions. Duke Math. J., 53(1), 249–269. MR: 87i:60078
(1988) Loop-erased self-avoiding random walk in two and three dimensions. J. Statist. Phys., 50(1–2), 91–108. MR: 89f:82053
(1991) Intersections of Random Walks. Birkhäuser, Boston. MR: 92f:60122
(2014) The probability that planar loop-erased random walk uses a given edge. Electron. Commun. Probab., 19, paper no. 51, 13. MR: 3246970
, , and (2004a) Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32(1B), 939–995. MR: 2044 671
, , and (2004b) On the scaling limit of planar self-avoiding walk. In and , 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
and (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
, , and (2002) Isoperimetric constants of infinite plane graphs. Discrete Comput. Geom., 28(3), 313–330. MR: 1923 955
(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
and (1998) Branching processes in Lévy processes: The exploration process. Ann. Probab., 26(1), 213–252. MR: 1617047
and (1991) The range of stable random walks. Ann. Probab., 19(2), 650–705. MR: 1106281
(1984) Frontiere de Poisson pour les groupes discrets de matrices. C. R. Acad. Sci. Paris Sér. I Math., 298(16), 393–396. MR: 748930
(1985) Poisson boundaries of discrete groups of matrices. Israel J. Math., 50(4), 319–336. MR: 800190
(1992) Sharp estimates for the entropy. In , editor, Proceedings of the International Meeting held in Frascati, July 1–10, 1991, pages 281–288. Plenum, New York. MR: 1222466
and (2013) Harmonic maps on amenable groups and a diffusive lower bound for random walks. Ann. Probab., 41(5), 3392–3419. MR: 3127886
, , and (2014) A Gaussian upper bound for martingale small-ball probabilities. Ann. Probab. To appear, http://www.arxiv.org/abs/1405.5980.
and (2012) A note on mixing times of planar random walks. Unpublished manuscript, http://www.arxiv.org/abs/1205.3980.
and (2010) Pólya's theorem on random walks via Pólya's urn. Amer. Math. Monthly, 117(3), 220–231. MR: 2640849
, , and (2009) Markov Chains and Mixing Times. American Mathematical Society, Providence, RI. With a chapter by and . MR: 2466937
(1995) On the cost of generating an equivalence relation. Ergodic Theory Dynam. Systems, 15(6), 1173–1181. MR: 96i:58091
(1985) An improved subadditive ergodic theorem. Ann. Probab., 13(4), 1279–1285. MR: 806224
(1987) Reversible growth models on symmetric sets. In . and , 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
, , and (1997) Domination by product measures. Ann. Probab., 25(1), 71–95. MR: 98f:60095
and (1996) On coupling of random walks and renewal processes. J. Appl. Probab., 33(1), 122–126. MR: 1371959
, , and (1995) The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2), 215–245. MR: 1337355
, , and (2002) Girth and Euclidean distortion. Geom. Funct. Anal., 12(2), 380–394. MR: 1911665
and (1949) An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc, 55, 961–962. MR: 0031538
and (2012) Diameter and spectral gap for planar graphs. Preprint, http://www.arxiv.org/abs/1204.4435.
and (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
and (2015) Cutoff on all Ramanujan graphs. Preprint, http://www.arxiv.org/abs/1507.04725.
, , and (1988) Ramanujan graphs. Combinatorica, 8(3), 261–277. MR: 963118
(2009) L2-invariants from the algebraic point of view. In , , and , 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
(1988) Strong laws of large numbers for weakly correlated random variables. Michigan Math. J., 35(3), 353–359. MR: 90d:60038
(1989) The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys., 125(2), 337–353. MR: 90h:82046
(1990) Random walks and percolation on trees. Ann. Probab., 18(3), 931–958. MR: 91i:60179
(1992) Random walks, capacity and percolation on trees. Ann. Probab., 20(4), 2043–2088. MR: 93k:60175
(1995) Random walks and the growth of groups. C. R. Acad. Sci. Paris Sér. I Math., 320(11), 1361–1366. MR: 96e:60015
(1996) Diffusions and random shadows in negatively curved manifolds. J. Funct. Anal., 138(2), 426–448. MR: 97d:58205
(2000) Phase transitions on nonamenable graphs. J. Math. Phys., 41(3), 1099–1126. MR: 2001c:82028
(2003) Determinantal probabilitymeasures. Publ.Math. Inst. Hautes Études Sci., 98(1), 167–212. MR: 2031202
(2005) Asymptotic enumeration of spanning trees. Combin. Probab. Comput., 14(4), 491–522. MR: 2160416
(2009) Random complexes and l2-Betti numbers. J. Topol. Anal., 1(2), 153–175. MR: 2541759
(2010) Identities and inequalities for tree entropy. Combin. Probab. Comput., 19(2), 303–313. MR: 2593624
(2013a) Distance covariance in metric spaces. Ann. Probab., 41(5), 3284–3305. MR: 3127883
(2013b) Fixed price of groups and percolation. Ergodic Theory Dynam. Systems, 33(1), 183–185. MR: 3009109
, , and (2008) Ends in uniform spanning forests. Electron. J. Probab., 13, paper no. 58, 1702–1725. MR: 2448128
and (2011) Perfect matchings as IID factors on non-amenable groups. European J. Combin., 32(7), 1115–1125. MR: 2825538
and (1992) Random walk in a random environment and first-passage percolation on trees. Ann. Probab., 20(1), 125–136. MR: 93c:60103
and (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
, , and (1995a) Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab., 23(3), 1125–1138. MR: 96m:60194
, , and (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
, , and (1996a) Biased random walks on Galton-Watson trees. Probab. Theory Related Fields, 106(2), 249–264. MR: 97h:60094
, , and (1996b) Random walks on the lamplighter group. Ann. Probab., 24(4), 1993–2006. MR: 97j:60014
, , and (1997) Unsolved problems concerning random walks on trees. In . and , 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
and (2015a) Cycle density in infinite Ramanujan graphs. Ann. Probab., 43(6), 3337–3358. MR: 3433583
and (2015b) Poisson boundaries of lamplighter groups: Proof of the Kaimanovich-Vershik conjecture. Preprint, http://www.arxiv.org/abs/1508.01845.
, , and (2003) Markov chain intersections and the loop-erased walk. Ann. Inst. H. Poincaré Probab. Statist., 39(5), 779–791. MR: 1997 212
, , and (2006) Minimal spanning forests. Ann. Probab., 34(5), 1665–1692. MR: 2271476
, , and (2008) Uniform non-amenability, cost, and the first l2-Betti number. Groups Geom. Dyn., 2(4), 595–617. MR: 2442947
and (1999) Indistinguishability of percolation clusters. Ann. Probab., 27(4), 1809–1836. MR: 1742 889
and (2003) Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J., 120(3), 515–575. MR: 2030095
and (2016) Invariant coupling of determinantal measures on sofic groups. Ergodic Theory Dynam. Systems, 36(2), 574–607.
(1983) A simple criterion for transience of a reversible Markov chain. Ann. Probab., 11(2), 393–402. MR: 84e:60102
(1987) Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differential Geom., 26(1), 33–66. MR: 892030
and (1994) Decomposition of Dirichlet processes and its application. Ann. Probab., 22(1), 494–524. MR: 1258888
and (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
(1935) Eine neue Definition der fastperiodischen Funktionen. Abh. Math. Semin. Hamb. Univ., 11, 240–244. http://dx.doi.org/10.1007/BF02940727.
(1982) Infinite distance transitive graphs of finite valency. Combinatorica, 2(1), 63–69. MR: 671146
(1970) Ü ber den Zusammenhang symmetrischer Graphen. Arch. Math. (Basel), 21, 331–336. MR: 44:6534
and (1993) The Self-Avoiding Walk. Birkhäuser, Boston. MR: 94f:82002
and (2014) Random walks on weakly hyperbolic groups. J. Reine Angew. Math. To appear, http://www.arxiv.org/abs/1410.4173.
and (2007a) Random walks on free products of cyclic groups. J. Lond. Math. Soc. (2), 75(1), 47–66. MR: 2302729
and (2007b) Randomly growing braid on three strands and the manta ray. Ann. Appl. Probab., 17(2), 502–536. MR: 2308334
(1992) Exact fractal dimension of the loop-erased self-avoiding random walk in two dimensions. Phys. Rev. Lett., 68, 2329–2331. http://dx.doi.org/10.1103/PhysRevLett.68.2329.
(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
, , and (2016) Boundaries of.n-free groups. In et al., editors, Groups, Graphs, and Random Walks, London Math. Soc. Lecture Notes Ser. Cambridge University Press. To appear, http://www.arxiv.org/abs/1211.3226.
and (2014) Poisson-Furstenberg boundaries of fundamental groups of closed 3-manifolds. Preprint, http://www.arxiv.org/abs/1403.2135.
(1982) The Fractal Geometry of Nature. W. H. Freeman, San Francisco. Schriftenreihe für den Referenten [Series for the Referee]. MR: 665254
(2012) How Groups Grow. Vol. 395 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge. MR: 2894945
(1998) The best bounds in a theorem of Russell Lyons. Electron. Comm. Probab., 3, paper no. 11, 91–94 (electronic). MR: 1650563
(2000) Loop-erased random walks, spanning trees and Hamiltonian cycles. Electron. Comm. Probab., 5, 39–50 (electronic). MR: 1736723
(2008) The lineage process in Galton-Watson trees and globally centered discrete snakes. Ann. Appl. Probab., 18(1), 209–244. MR: 2380897
and (2003) Ladder variables, internal structure of Galton-Watson trees and finite branching random walks. J. Appl. Probab., 40(3), 671–689. MR: 1993260
(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
and (2013) Locality of percolation for abelian Cayley graphs. Ann. Probab. To appear, http://www.arxiv.org/abs/1312.1946.
(1991) A Basic Course in Algebraic Topology. Springer-Verlag, New York. MR: 92c:55001
(2009) The growth exponent for planar loop-erased random walk. Electron. J. Probab., 14, paper no. 36, 1012–1073. MR: 2506124
(2002) Lectures on Discrete Geometry. Vol. 212 of Graduate Texts in Mathematics. Springer-Verlag, New York. MR: 1899299
(2008) On variants of the Johnson-Lindenstrauss lemma. Random Structures Algorithms, 33(2), 142–156. MR: 2436844
and (1998) Hyperbolic Manifolds and Kleinian Groups. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York. MR: 1638795
(1988) Covering problems for Brownian motion on spheres. Ann. Probab., 16(1), 189–199. MR: 920264
(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
and (1986) Random recursive constructions: Asymptotic geometric and topological properties. Trans. Amer. Math. Soc., 295(1), 325–346. MR: 87j:60027
and (1940) Random paths in two and three dimensions. Proc. Roy. Soc. Edinburgh, 60, 281–298. MR: 2,107f
(1989) On the method of bounded differences. In , 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
(1989) Random Walks on Graphs and Directed Graphs. Ph.D. thesis, University of Waterloo.
and (1995) Extension of Foster's averaging formula to infinite networks with moderate growth. Math. Z., 219(2), 171–185. MR: 96g:94031
and (1970) The distance between points in random trees. J. Combinatorial Theory, 8, 99–103. MR: 0263685
and (2009) Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab., 37(5), 1679– 1714. MR: 2561431
(2013) Invariant monotone coupling need not exist. Ann. Probab., 41(3A), 1180–1190. MR: 3098675
(1988) Nonnegative Matrices.Wiley-Interscience Series in Discrete Mathematics and Optimization. JohnWiley, New York. MR: 89i:15001
(1988) Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra Appl., 103, 119–131. MR: 89k:05071
(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
(1964) Lattice statistics. In , editor, Applied Combinatorial Mathematics, pages 96–143. John Wiley, New York. University of California Engineering and Physical Sciences Extension Series. MR: 30:4687
(1967) Various proofs of Cayley's formula for counting trees. In and , editors, A Seminar on Graph Theory, pages 70–78. Holt, Rinehart and Winston, New York. MR: 35:5365
(1954) A note on unramified abelian covering surfaces of a closed Riemann surface. J. Math. Soc. Japan, 6, 162–176. MR: 0066468
(2003) The components of the wired spanning forest are recurrent. Probab. Theory Related Fields, 125(2), 259–265. MR: 1961 344
and (2005) Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields, 133(2), 245–266. MR: 2198701
and (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
and (2001) Percolation on Grigorchuk groups. Comm. Algebra, 29(2), 661–671. MR: 2002e:82033
(2003) Ramanujan graphs. J. Ramanujan Math. Soc., 18(1), 33–52. MR: 1966527
(1965) The Haar Integral. D. Van Nostrand, Princeton, NJ. MR: 0175995
(2010) L1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry. In , , , , and , editors, Proceedings of the International Congress of Mathematicians. Volume III, pages 1549–1575. Hindustan Book Agency, New Delhi. MR: 2827855
and (2008) Embeddings of discrete groups and the speed of random walks. Int. Math. Res. Not. IMRN, 2008, Art. rnn 076, 34. MR: 2439557
, , , and (2006) Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1), 165–197. MR: 2239346
(1959) Random walk and electric currents in networks. Proc. Cambridge Philos. Soc., 55, 181–194. MR: 23:A2239
(1961) Edge-disjoint spanning trees of finite graphs. J. London Math. Soc., 36, 445–450. MR: 0133253
and (2011) On the inverse mean first passage matrix problem and the inverse M-matrix problem. Linear Algebra Appl., 434(7), 1620–1630. MR: 2775741
(1986) Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist., 22(2), 199–207. MR: 88a:60150
and (2013) The Poisson boundary of CAT??0_ cube complex groups. Groups Geom. Dyn., 7(3), 653–695. MR: 3095714
(1997) Topics in Disordered Systems. Birkhäuser, Basel. MR: 99e:82052
and (1981) Infinite clusters in percolation models. J. Statist. Phys., 26(3), 613–628. MR: 83e:82038
and (1996) Ground-state structure in a highly disordered spin-glass model. J. Statist. Phys., 82(3–4), 1113–1132. MR: 97a:82054
and (2006) Short-range spin glasses: Selected open problems. In , , , , and , 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
(1982) Exact critical point and critical exponents of On models in two dimensions. Phys. Rev. Lett., 49(15), 1062–1065. http://dx.doi.org/10.1103/PhysRevLett.49.1062.
(1991) On the second eigenvalue of a graph. Discrete Math., 91(2), 207–210. MR: 1124768
(2004) Tight estimates for eigenvalues of regular graphs. Electron. J. Combin., 11(1), Note 9, 4 pp. (electronic). MR: 2056091
, , and (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. http://dl.acm.org/citation.cfm?id=2722129.2722247.
(1992) Cogrowth of regular graphs. Proc. Amer. Math. Soc., 116(1), 203–205. MR: 1120509
and (1987) Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math., 48, 1–141. MR: 88j:28014
and (1970) An operator theorem on L1 convergence to zero with applications to Markov kernels. Ann. Math. Statist., 41, 1631–1639. MR: 0272057
(2015) A functional analysis proof of Gromov's polynomial growth theorem. Preprint, http://www.arxiv.org/abs/1510.04223.
and (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
and (1991) On family trees and subtrees of simple branching processes. J. Theoret. Probab., 4(2), 353–369. MR: 92f:60145
and (1992) Length-biasing, characterizations of laws and the moment problem. Austral. J. Statist., 34(2), 307–322. MR: 94a:60018
A.C. and (1932) A note on analytic functions in the unit circle. Proc. Camb. Phil. Soc., 28, 266–272. http://dx.doi.org/10.1017/S0305004100010112.
(1996) How to cheat a bad mathematician. EEC HC&M Network on Complexity and Chaos. #ERBCHRX- CT940546. ISI, Torino, Italy. Unpublished. Available at http://seneca.fis.ucm.es/parr/GAMES/cheat.pdf.
(1988) Amenability. American Mathematical Society, Providence, RI. MR: 90e:43001
and (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
(1936) On Ising's model of ferromagnetism. Math. Proc. Cambridge Philos. Soc., 32, 477–481. http://dx.doi.org/10.1017/S0305004100019174.
and (2011) New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab., 39(2), 587–608. MR: 2789507
(1988) Phase transition in reinforced random walk and RWRE on trees. Ann. Probab., 16(3), 1229–1241. MR: 89g:60220
(1991) Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., 19(4), 1559–1574. MR: 92g:60014
(2000) Towards a theory of negative dependence. J. Math. Phys., 41(3), 1371–1390. MR: 2001g:62039
and (1995a) Critical random walk in random environment on trees. Ann. Probab., 23(1), 105–140. MR: 96f:60123
and (1995b) Galton-Watson trees with the same mean have the same polar sets. Ann. Probab., 23(3), 1102–1124. MR: 96i:60093
and (1996) On which graphs are all random walks in random environments transient? In and , 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
and (2000) Nonamenable products are not treeable. Israel J. Math., 118, 147–155. MR: 2001j:43002
and (2010) The critical Ising model on trees, concave recursions and nonlinear capacity. Ann. Probab., 38(1), 184–206. MR: 2599197
and (2001) The branching random walk and contact process on Galton-Watson and nonhomogeneous trees. Ann. Probab., 29(4), 1563–1590. MR: 1880232
(1996) Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys., 177(2), 417–434. MR: 98k:60143
(1999) Probability on trees: An introductory climb. In , 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
(2000) Percolation on nonamenable products at the uniqueness threshold. Ann. Inst. H. Poincaré Probab. Statist., 36(3), 395–406. MR: 2001f:60114
, , and (2006) Critical percolation on certain nonunimodular graphs. New York J. Math., 12, 1–18 (electronic). MR: 2217160
, , and (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
and (1998) The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields, 111(1), 141–165. MR: 99e:60217
and (2008) A central limit theorem for biased random walks on Galton-Watson trees. Probab. Theory Related Fields, 140(3–4), 595–629. MR: 2365486
(2001) Probability Theory on Galton-Watson Trees. Ph.D. thesis, Massachusetts Institute of Technology. Available at http://hdl.handle.net/1721.1/8673.
(1997) Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics. University of Chicago Press, Chicago. MR: 1489237
(2008) A note on percolation on.d: Isoperimetric profile via exponential cluster repulsion. Electron. Commun. Probab., 13, 377–392. MR: 2415145
(1983) Ergodic Theory. Cambridge University Press, Cambridge. MR: 87i:28002
(2008) A probabilistic approach to Carne's bound. Potential Anal., 29(1), 17–36. MR: 2421492
(1996) Functional limit theorems for the simple random walk on a supercritical Galton-Watson tree. In , , and , 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
(1998) Théoreme central limite fonctionnel pour une marche au hasard en environnement aléatoire. Ann. Probab., 26(3), 1016–1040. MR: 1634413
(2002) Scaling exponents of random walks in random sceneries. Stochastic Process. Appl., 100, 3–25. MR: 1919605
(1973) On the complexity of a concentrator. In Proceedings of the Seventh International Teletraffic Congress (Stockholm, 1973), pages 318/1–318/4.
(1993) Probability. Springer, New York. http://dx.doi.org/10.1007/978-1-4612-4374-8.
(1998) Enumerations of trees and forests related to branching processes and random walks. In and , 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
(1982) Positively correlated normal variables are associated. Ann. Probab., 10(2), 496–499. MR: 665603
and (1999) Amenable groups, isoperimetric profiles and random walks. In Geometric Group Theory Down Under (Canberra, 1996), pages 293–316. de Gruyter, Berlin. MR: 1714851
and (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 http://www.math.cornell.edu/_lsc/surv.ps.gz.
, , and (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. http://iopscience.iop.org/1742-5468/2011/10/P10004.
(1921) Ü ber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strasennetz. Math. Ann., 84(1–2), 149–160. MR: 1512028
(1998) The combinatorics of effective resistances and resistive inverses. Inform. and Comput., 147(2), 209–223. MR: 1662276
and (1978) Brownian Motion and Classical Potential Theory. Probability and Mathematical Statistics. Academic Press [Harcourt Brace Jovanovich Publishers], New York. MR: 0492329
and (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
(2015) Expansion of random graphs: New proofs, new results. Invent. Math., 201(3), 845–908. MR: 3385636
(2003) On random walks and diffusions related to Parrondo's games. In , , and , 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
(2006) Foundations of Hyperbolic Manifolds, 2nd ed. Vol. 149 of Graduate Texts in Mathematics. Springer, New York. MR: 2249478
(2004) A general Choquet-Deny theorem for nilpotent groups. Ann. Inst. H. Poincaré Probab. Statist., 40(6), 677–683. MR: 2096214
(2000) Proof of the van den Berg-Kesten conjecture. Combin. Probab. Comput., 9(1), 27–32. MR: 1751301
(2001) Biased random walk on lamplighter groups and graphs. J. Theoret. Probab., 14(2), 379–391. MR: 1838 734
(2005) Random Walk in Random and Non-Random Environments, 2nd ed. World Scientific, Hackensack, NJ. MR: 2168855
and (2007) The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN, 2007, Art. rnm006, 33. MR: 2361453
(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
(1981) Ergodic and mixing random walks on locally compact groups. Math. Ann., 257(1), 31–42. MR: 630645
(1971) Markov Processes. Structure and Asymptotic Behavior. Vol. 184 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York. MR: 48:7379
(1996) Stochastic Processes, 2nd ed., John Wiley, New York. MR: 97a:60002
and (2007) A Second Course in Probability. ProbabilityBookstore.com, Boston.
(1952) Harmonic functions on open Riemann surfaces. Trans. Amer. Math. Soc., 73, 40–94. MR: 0049396
(1988) Real Analysis, 3rd ed. Macmillan, New York. MR: 90g:00004
(1987) Real and Complex Analysis, 3rd ed. McGraw-Hill, New York. MR: 88k:00002
(1991) Functional Analysis, 2nd ed. International Series in Pure and Applied Mathematics. McGraw-Hill, New York. MR: 92k:46001
(1964) Vertex-transitive graphs. Monatsh. Math., 68, 426–438. MR: 0175815
and (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
(1995) Isoperimetric inequalities and decay of iterated kernels for almost-transitive Markov chains. Combin. Probab. Comput., 4(4), 419–442. MR: 1377559
(1992) On the norms of group-invariant transition operators on graphs. J. Theoret. Probab., 5(3), 563–576. MR: 93h:60113
(2010a) Lamplighter Random Walks and Entropy-Sensitivity of Languages. Ph.D. thesis, Technische Universität Graz. Available at http://www.arxiv.org/abs/1012.2757.
(2010b) A note on the Poisson boundary of lamplighter random walks. Monatsh. Math., 159(4), 379–396. MR: 2600904
(1997) Martin boundaries and random walks. In , 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
(1979) Polynomidentitäten und Permutationsdarstellungen lokalkompakter Gruppen. Invent. Math., 55(2), 97–106. MR: 81d:22006
(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
(1938) Metric spaces and positive definite functions. Trans. Amer. Math. Soc., 44(3), 522–536. MR: 1501980
(1992) On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab., 20(1), 174–193. MR: 1143417
(1999a) Percolation in 1 + 1 dimensions at the uniqueness threshold. In and , editors, Perplexing Problems in Probability, pages 53–67. Birkhäuser, Boston. Festschrift in honor of Harry Kesten. MR: 1703 124
(1999b) Stability of infinite clusters in supercritical percolation. Probab. Theory Related Fields, 113(2), 287–300. MR: 1676831
(2001) Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys., 219(2), 271–322. MR: 2002h:82036
(2000) Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118, 221–288. MR: 1776 084
(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
(1968) On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist., 39, 2098–2102. MR: 38:2847
(1970) On the supercritical Galton-Watson process with immigration. Math. Biosci., 7, 9–14. MR: 42:5348
(2007) Gaussian free fields for mathematicians. Probab. Theory Related Fields, 139(3–4), 521–541. MR: 2322706
(1972) Covering the circle with random arcs. Israel J. Math., 11, 328–345. MR: 45:4468
(1987) The ergodic and entropy theorems revisited. IEEE Trans. Inform. Theory, 33(2), 263–266. MR: 880168
and (2000) Spanning trees on graphs and lattices in d dimensions. J. Phys. A, 33(21), 3881–3902. MR: 2001b:05111
and (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. http://dx.doi.org/10.1145/2591796.2591803.
(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
(2010) Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2), 172(2), 1435–1467. MR: 2680496
(1993) Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. Amer. Math. Soc., 119(4), 1239–1248. MR: 94a:31004
and (1990) Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z., 205(3), 471–486. MR: 91m:43002
(1975) Random walks in a random environment. Ann. Probab., 3, 1–31. MR: 50:14943? Spakulová, I.
(2009) Critical percolation of virtually free groups and other tree-like graphs. Ann. Probab., 37(6), 2262–2296. MR: 2573558
and (2007) Spectral partitioning works: Planar graphs and finite element meshes. Linear Algebra Appl., 421(2–3), 284–305. MR: 2294342
(1962) Hitting probabilities. J. Math. Mech., 11, 593–614. MR: 0139219
(1964) Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 3, 110–121. MR: 0172343
(1976) Principles of Random Walk, 2nd ed. Vol. 34 of Graduate Texts in Mathematics. Springer-Verlag, New York. MR: 52:9383
(1996) The existence of an intermediate phase for the contact process on trees. Ann. Probab., 24(4), 1711–1726. MR: 97j:60191
(1966) Operator limit theorems. Trans. Amer. Math. Soc., 121, 90–115. MR: 0190757
(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
(1965) The existence of probability measures with given marginals. Ann. Math. Statist., 36, 423–439. MR: 31:1693
(2009) The contact process seen from a typical infected site. J. Theoret. Probab., 22(3), 711–740. MR: 2530110
and (2005a) Hierarchical clustering via joint between-within distances: Extending Ward's minimum variance method. J. Classification, 22(2), 151–183. MR: 2231170
and (2005b) A new test for multivariate normality. J. Multivariate Anal., 93(1), 58–80. MR: 2119764
, , and (2007) Measuring and testing dependence by correlation of distances. Ann. Statist., 35(6), 2769–2794. MR: 2382665
(1998) Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics. Springer-Verlag, Berlin. MR: 1717054
(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
(1997) Random walk on periodic trees. Electron. J. Probab., 2, paper no. 1, 1–16 (electronic). MR: 1436761
(1998) Biased random walks on directed trees. Probab. Theory Related Fields, 111(1), 123–139. MR: 1626778
(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
and (1997) A note on distributional equality in the cyclic tour property for Markov chains. Combin. Probab. Comput., 6(4), 493–496. MR: 1483432
and (1980) Introduction to Functional Analysis, 2nd ed. John Wiley, New York. MR: 564653
(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
(1974) Enumeration of graphs on a large periodic lattice. In and , 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
(1991) Random walks and the effective resistance of networks. J. Theoret. Probab., 4(1), 101–109. MR: 92c:60097
(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
(1994b) An extension of Foster's network theorem. Combin. Probab. Comput., 3(3), 421–427. MR: 1300977
(2015) A remark about the spectral radius. Int. Math. Res. Not. IMRN, 2015(10), 2856–2864. MR: 3352259
(2016) The expected degree of minimal spanning forests. Combinatorica. Online http://dx.doi.org/10.1007/s00493-014-3160-x.
(1989) Transient random walks, harmonic functions, and electrical currents in infinite electrical networks. Technical Report Mat-Report n. 1989-07, Technical University of Denmark.
(1990) Resistances and currents in infinite electrical networks. J. Combin. Theory Ser. B, 49(1), 87–102. MR: 91d:94029
(1992) Isoperimetric inequalities and transient random walks on graphs. Ann. Probab., 20(3), 1592–1600. MR: 1175279
(2000) Coupling, Stationarity, and Regeneration. Probability and Its Applications. Springer-Verlag, New York. MR: 1741181
(2006a) Ends in free minimal spanning forests. Ann. Probab., 34(3), 865–869. MR: 2243871
(2006b) Neighboring clusters in Bernoulli percolation. Ann. Probab., 34(6), 2332–2343. MR: 2294984
(2006c) Percolation on nonunimodular transitive graphs. Ann. Probab., 34(6), 2344–2364. MR: 2294985
(2007) Cutsets in infinite graphs. Combin. Probab. Comput., 16(1), 159–166. MR: 2286517
(2015) Indistinguishability of components of random spanning forests. Preprint, http://www.arxiv.org/abs/1506.01370.
(1991) Block designs and electrical networks. Ann. Statist., 19(2), 1010–1027. MR: 1105858
(1988) Which sets contain multiple points of Brownian motion?, Math. Proc. Cambridge Philos. Soc., 103(1), 181–187. MR: 913461
(1982) Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc., 91(1), 57–74. MR: 633256
(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
(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
(1989) On the delta-wye reduction for planar graphs. J. Graph Theory, 13(2), 141–148. MR: 90c:05078
(1961) On the problem of decomposing a graph into n connected factors. J. London Math. Soc., 36, 221–230. MR: 0140438
(1998) Green's functions for random walks on ZN. Proc. London Math. Soc. (3), 77(1), 215–240. MR: 1625467
(1985a) Isoperimetric inequalities and Markov chains. J. Funct. Anal., 63(2), 215–239. MR: 87e:60124
(1985b) Long range estimates for Markov chains. Bull. Sci. Math. (2), 109(3), 225–252. MR: 87j:60100
(1986) Théorie du potentiel sur des groupes et des variétés. C. R. Acad. Sci. Paris Sér. I Math., 302(6), 203–205. MR: 832044
and (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
and (1993) Branching processes. II. J. Soviet Math., 67(6), 3407–3485. MR: 1260986
and (1979) Random walks on groups: Boundary, entropy, uniform distribution. Dokl. Akad. Nauk SSSR, 249(1), 15–18. MR: 553972
(2000a) Anchored expansion and random walk. Geom. Funct. Anal., 10(6), 1588–1605. MR: 1810 755
(2000b) On the speed of random walks on graphs. Ann. Probab., 28(1), 379–394. MR: 2001g:60173
(2002) Fast graphs for the random walker. Probab. Theory Related Fields, 124(1), 50–72. MR: 1929811
(2001) Vertex-reinforced random walk on arbitrary graphs. Ann. Probab., 29(1), 66–91. MR: 1825142
(1982) An Introduction to Ergodic Theory. Springer-Verlag, New York. MR: 84e:28017
(1970) Connectivity of transitive graphs. J. Combinatorial Theory, 8, 23–29. MR: 0266804
and (1996) A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc., 348(2), 585–632. MR: 1322959
(1964) Some Strong Laws for Random Walks and Brownian Motion. Ph.D. thesis, Cornell University. MR: 2614450
(1932) Congruent graphs and the connectivity of graphs. Amer. J. Math., 54(1), 150–168. MR: 1506881
(1999) Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture. Duke Math. J., 99(1), 93–112. MR: 1700742
and (2003) Winding angle variance of Fortuin-Kasteleyn contours. Phys. Rev. E, 68, 056101. http://dx.doi.org/10.1103/PhysRevE.68.056101.
and (1983) Invasion percolation: A new form of percolation theory. J. Phys. A, 16(14), 3365–3376. MR: 725616
(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
(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
(2004a) Red-green-blue model. Phys. Rev. E (3), 69, 037105. http://dx.doi.org/10.1103/PhysRevE.69.037105.
(2004b) On exponential growth and uniformly exponential growth for groups. Invent. Math., 155(2), 287–303. MR: 2031429
(2004c) Further groups that do not have uniformly exponential growth. J. Algebra, 279(1), 292–301. MR: 2078400
(1969) Scoring rules and the evaluation of probability assessors. J. Amer. Stat. Assoc., 64(327), 1073–1078. http://dx.doi.org/10.1007/BF02562681.
(1991) Topological groups and infinite graphs. Discrete Math., 95(1–3), 373–384. MR: 93i:22004
(2000) RandomWalks on Infinite Graphs and Groups. Vol. 138 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge. MR: 2001k:60006
(1968) Growth of finitely generated solvable groups and curvature of Riemanniann manifolds. J. Differential Geometry, 2, 421–446. MR: 0248688
(1947) Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.), 56, 795–798. MR: 9,149e
(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
(1976) Infinite electrical networks. Proc. IEEE, 64(1), 6–17. Recent trends in system theory. MR: 0453371
(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
(2011) 70+ years of the Watson integrals. J. Stat. Phys., 145(3), 591–612. MR: 2862945
(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