Abért, M., Glasner, Y., and Virág, B. (2016) The measurable Kesten theorem. Ann. Probab., 44(3), 1601–1646.Google Scholar

Abrams, A. and Kenyon, R. (2015) Fixed-energy harmonic functions. Preprint, :http://www.arxiv.org/abs/1505.05785.

Achlioptas, D. and McSherry, F. (2001) Fast computation of low rank matrix approximations. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pages 611–618 (electronic). ACM, New York. Held in Hersonissos, 2001, available electronically at http://portal.acm.org/toc.cfm?id=380752. MR: 2120364

Achlioptas, D. and McSherry, F. (2007) Fast computation of low-rank matrix approximations. J. ACM, 54(2), Art. 9. MR: 2295993Google Scholar

Adams, S. (1988) Indecomposability of treed equivalence relations. Israel J. Math., 64(3), 362–380. MR: 995576Google Scholar

Adams, S. (1990) Trees and amenable equivalence relations. Ergodic Theory Dynam. Systems, 10(1), 1–14. MR: 91d:28041Google Scholar

Adams, S. and Lyons, R. (1991) Amenability, Kazhdan's property and percolation for trees, groups and equivalence relations. Israel J. Math., 75(2–3), 341–370. MR: 93j:43001Google Scholar

Adams, S. and Spatzier, R.J. (1990) Kazhdan groups, cocycles and trees. Amer. J. Math., 112(2), 271–287. MR: 91c:22011Google Scholar

Adler, J. and Lev, U. (2003) Bootstrap percolation: Visualizations and applications. Brazilian J. Phys., 33(3), 641–644. http://dx.doi.org/10.1590/S0103-97332003000300031.Google Scholar

Aïdékon, E. (2008) Transient random walks in random environment on a Galton-Watson tree. Probab. Theory Related Fields, 142(3–4), 525–559. MR: 2438700Google Scholar

Aïdékon, E. (2010) Large deviations for transient random walks in random environment on a Galton-Watson tree. Ann. Inst. Henri Poincaré Probab. Stat., 46(1), 159–189. MR: 2641775Google Scholar

Aïdékon, E. (2011) Uniform measure on a Galton-Watson tree without the X log X condition. Preprint, http://www.arxiv.org/abs/1101.1816.

Aïdékon, E. (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.

Aïdékon, E. (2014) Speed of the biased random walk on a Galton-Watson tree. Probab. Theory Related Fields, 159(3–4), 597–617. MR: 3230003Google Scholar

Aizenman, M. (1985) The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory. Comm. Math. Phys., 97(1–2), 91–110. MR: 782960Google Scholar

Aizenman, M. and Barsky, D.J. (1987) Sharpness of the phase transition in percolation models. Comm. Math. Phys., 108(3), 489–526. MR: 88c:82026Google Scholar

Aizenman, M., Burchard, A., Newman, C.M., and Wilson, D.B. (1999) Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms, 15(3–4), 319–367. MR: 2001c:60151Google Scholar

Aizenman, M., Chayes, J.T., Chayes, L., and Newman, C.M. (1988) Discontinuity of the magnetization in one-dimensional Ising and Potts models. J. Statist. Phys., 50(1–2), 1–40. MR: 89f:82072Google Scholar

Aizenman, M., Kesten, H., and Newman, C.M. (1987) Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys., 111(4), 505–531. MR: 89b:82060Google Scholar

Aizenman, M. and Lebowitz, J.L. (1988) Metastability effects in bootstrap percolation. J. Phys. A, 21(19), 3801–3813. MR: 968311Google Scholar

Aldous, D.J (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.Google Scholar

Aldous, D.J (1990) The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math., 3(4), 450–465. MR: 91h:60013Google Scholar

Aldous, D.J (1991) Random walk covering of some special trees. J. Math. Anal. Appl., 157(1), 271–283. MR: 1109456Google Scholar

Aldous, D.J and Fill, J.A. (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.

Aldous, D.J and Lyons, R. (2007) Processes on unimodular random networks. Electron. J. Probab., 12, paper no. 54, 1454–1508 (electronic). MR: 2354165Google Scholar

Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., and Rackoff, C. (1979) Random walks, universal traversal sequences, and the complexity of maze problems. In 20th Annual Symposium on Foundations of Computer Science, pages 218–223. IEEE, New York. Held in San Juan, Puerto Rico, October 29–31, 1979. MR: 598110

Alexander, K.S (1995a) Percolation and minimal spanning forests in infinite graphs. Ann. Probab., 23(1), 87–104. MR: 96c:60114Google Scholar

Alexander, K.S (1995b) Simultaneous uniqueness of infinite clusters in stationary random labeled graphs. Comm. Math. Phys., 168(1), 39–55. Erratum: Comm. Math. Phys. 172, (1995), 221. MR: 96e:60166aGoogle Scholar

Alexander, K.S and Molchanov, S.A. (1994) Percolation of level sets for two-dimensional random fields with lattice symmetry. J. Statist. Phys., 77(3–4), 627–643. MR: 95i:82052Google Scholar

Alon, N. (1986) Eigenvalues and expanders. Combinatorica, 6(2), 83–96. MR: 88e:05077Google Scholar

Alon, N. (2003) Problems and results in extremal combinatorics. I. Discrete Math., 273(1–3), 31–53. MR: 2025940Google Scholar

Alon, N., Benjamini, I., Lubetzky, E., and Sodin, S. (2007) Non-backtracking random walks mix faster. Commun. Contemp. Math., 9(4), 585–603. MR: 2348845Google Scholar

Alon, N., Benjamini, I., and Stacey, A. (2004) Percolation on finite graphs and isoperimetric inequalities. Ann. Probab., 32(3A), 1727–1745. MR: 2073175Google Scholar

Alon, N., Hoory, S., and Linial, N. (2002) The Moore bound for irregular graphs. Graphs Combin., 18(1), 53–57. MR: 1892433Google Scholar

Alon, N. and Milman, V.D. (1985) isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B, 38(1), 73–88. MR: 782626Google Scholar

Amir, G. and Virág, B. (2016) Speed exponents of random walks on groups. IMRN, 2016. http://dx.doi.org/10.1093/imrn/rnv378.Google Scholar

Anantharam, V. and Tsoucas, P. (1989) A proof of the Markov chain tree theorem. Statist. Probab. Lett., 8(2), 189–192. MR: 1017890Google Scholar

Ancona, A. (1988) Positive harmonic functions and hyperbolicity. In Král, J., Lukes, J., Netuka, I., and Vesely, J., editors, Potential Theory—Surveys and Problems (Prague, 1987), pages 1–23. Springer, Berlin. MR: 973 878

Ancona, A., Lyons, R., and Peres, Y. (1999) Crossing estimates and convergence of Dirichlet functions along random walk and diffusion paths. Ann. Probab., 27(2), 970–989. MR: 1698991Google Scholar

Angel, O., Barlow, M.T., Gurel-Gurevich, O., and Nachmias, A. (2016) Boundaries of planar graphs, via circle packings. Ann. Probab., 44(3), 1956–1984. MR: 3502598Google Scholar

Angel, O. and Benjamini, I. (2007) A phase transition for the metric distortion of percolation on the hypercube. Combinatorica, 27(6), 645–658. MR: 2384409Google Scholar

Angel, O., Benjamini, I., Berger, N., and Peres, Y. (2006) Transience of percolation clusters on wedges. Electron. J. Probab., 11, paper no. 25, 655–669 (electronic). MR: 2242658Google Scholar

Angel, O., Crawford, N., and Kozma, G. (2014) Localization for linearly edge reinforced random walks. Duke Math. J., 163(5), 889–921. MR: 3189433Google Scholar

Angel, O., Friedman, J., and Hoory, S. (2015) The non-backtracking spectrum of the universal cover of a graph. Trans. Amer. Math. Soc., 367(6), 4287–4318. MR: 3324928Google Scholar

Angel, O., Goodman, J., den Hollander, F., and Slade, G. (2008) Invasion percolation on regular trees. Ann. Probab., 36(2), 420–466. MR: 2393988Google Scholar

Angel, O., Goodman, J., and Merle, M. (2013) Scaling limit of the invasion percolation cluster on a regular tree. Ann. Probab., 41(1), 229–261. MR: 3059198Google Scholar

Angel, O. and Szegedy, B. (2016) Recurrence of weak limits of excluded minor graphs. In preparation.

Arratia, R. and Goldstein, L. (2010) Size bias, sampling, the waiting time paradox, and infinite divisibility: When is the increment indepen-dent? Preprint, http://www.arxiv.org/abs/1007.3910.

Asadpour, A., Goemans, M.X., M'adry, A., Oveis Gharan, S., and Saberi, A. (2010) An n-approximation algorithm for the asymmetric traveling salesman problem. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 379–389. SIAM, Philadelphia. Held in Austin, TX, January 17–19, 2010. MR: 2809683

Asmussen, S. and Hering, H. (1983) Branching Processes. Birkhäuser, Boston. MR: 85b:60076

Athreya, K.B (1971) A note on a functional equation arising in Galton-Watson branching processes. J. Appl. Probability, 8, 589–598. MR: 45:1271Google Scholar

Athreya, K.B and Ney, P.E. (1972) Branching Processes. Vol. 196 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York. MR: 51:9242

Atiyah, M.F (1976) Elliptic operators, discrete groups and von Neumann algebras. In Colloque “Analyse et Topologie” en l'Honneur de Henri Cartan, pages 43–72. Astérisque, 32–33. Soc. Math. France, Paris. Tenu le 17–20 juin 1974 à Orsay. MR: 0420729

Avez, A. (1972) Entropie des groupes de type fini. C. R. Acad. Sci. Paris Sér. A-B, 275, A1363–A1366. MR: 0324741Google Scholar

Avez, A. (1974) Théorème de Choquet-Deny pour les groupes à croissance non exponentielle. C. R. Acad. Sci. Paris Sér. A, 279, 25–28. MR: 0353405Google Scholar

Avez, A. (1976) Croissance des groupes de type fini et fonctions harmoniques. In Théorie Ergodique, Lecture Notes in Mathematics, Vol. 532, pages 35–49. Springer, Berlin. Actes des Journées Ergodiques, Rennes, 1973/1974, Edité par J.-P. Conze et M. S. Keane. MR: 0482911

Azuma, K. (1967) Weighted sums of certain dependent random variables. Tohoku Math. J. (2), 19, 357–367. MR: 0221571Google Scholar

Babai, L. (1991) Local expansion of vertex-transitive graphs and random generation in finite groups. In STOC '91: Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, pages 164–167. ACM, New York. http://dx.doi.org/10.1145/103418.103440.

Babai, L. (1997) The growth rate of vertex-transitive planar graphs. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 564–573. ACM, New York. Held in New Orleans, LA, January 5–7, 1997. MR: 1447704

Babson, E. and Benjamini, I. (1999) Cut sets and normed cohomology with applications to percolation. Proc. Amer. Math. Soc., 127(2), 589–597. MR: 99g:05119Google Scholar

Bahar, I., Atilgan, A.R., and Erman, B. (1997) Direct evaluation of thermal fluctuations in protein using a single parameter harmonic potential. Folding & Design, 2, 173–181. http://dx.doi.org/10.1016/S1359-0278(97)00024-2.Google Scholar

Bakirov, N.K, Rizzo, M.L., and Székely, G.J. (2006) A multivariate nonparametric test of independence. J. Multivariate Anal., 97(8), 1742–1756. MR: 2298886Google Scholar

Balázs, M. and Folly, A. (2014) Electric network for non-reversible Markov chains. Preprint, http://www.arxiv.org/abs/1405.

Ball, K. (1992) Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal., 2(2), 137–172. MR: 1159828Google Scholar

Ballmann, W. and Ledrappier, F. (1994) The Poisson boundary for rank one manifolds and their cocompact lattices. Forum Math., 6(3), 301–313. MR: 1269841Google Scholar

Balogh, J., Bollobás, B., Duminil-Copin, H., and Morris, R. (2012) The sharp threshold for bootstrap percolation in all dimensions. Trans. Amer. Math. Soc., 364(5), 2667–2701. MR: 2888224Google Scholar

Balogh, J., Bollobás, B., and Morris, R. (2009) Bootstrap percolation in three dimensions. Ann. Probab., 37(4), 1329–1380. MR: 2546747Google Scholar

Balogh, J., Peres, Y., and Pete, G. (2006) Bootstrap percolation on infinite trees and non-amenable groups. Combin. Probab. Comput., 15(5), 715–730. MR: 2248323Google Scholar

Barlow, M.T (2016) Loop erased walks and uniform spanning trees. In Discrete Geometric Analysis, vol. 34 of MSJ Memoirs, pages 1–32. Mathematical Society of Japan, Tokyo.

Barlow, M.T and Masson, R. (2010) Exponential tail bounds for loop-erased random walk in two dimensions. Ann. Probab., 38(6), 2379–2417. MR: 2683633Google Scholar

Barlow, M.T and Perkins, E.A. (1989) Symmetric Markov chains in Zd: How fast can they move? Probab. Theory Related Fields, 82(1), 95–108. MR: 997432Google Scholar

Barreira, L. (2008) Dimension and Recurrence in Hyperbolic Dynamics. Vol. 272 of Progress in Mathematics. Birkhäuser, Basel. MR: 2434246

Barsky, D.J, Grimmett, G.R., and Newman, C.M. (1991) Percolation in half-spaces: Equality of critical densities and continuity of the percolation probability. Probab. Theory Related Fields, 90(1), 111–148. MR: 92m:60086Google Scholar

Bartholdi, L. (1999) Counting paths in graphs. Enseign. Math. (2), 45(1–2), 83–131. MR: 1703364Google Scholar

Bartholdi, L. (2003) A Wilson group of non-uniformly exponential growth. C. R. Math. Acad. Sci. Paris, 336(7), 549–554. MR: 1981466Google Scholar

Bartholdi, L., Kaimanovich, V.A., and Nekrashevych, V.V. (2010) On amenability of automata groups. Duke Math. J., 154(3), 575–598. MR: 2730578Google Scholar

Bartholdi, L. and Virág, B. (2005) Amenability via random walks. Duke Math. J., 130(1), 39–56. MR: 2176547Google Scholar

Bass, R.F (1995) Probabilistic Techniques in Analysis. Probability and Its Applications. Springer-Verlag, New York. MR: 96e:60001

Bateman, M. and Katz, N.H. (2008) Kakeya sets in Cantor directions. Math. Res. Lett., 15(1), 73–81. MR: 2367175Google Scholar

Beardon, A.F and Stephenson, K. (1990) The uniformization theorem for circle packings. Indiana Univ. Math. J., 39(4), 1383–1425. MR: 1087197Google Scholar

Bekka, M.EB.and Valette, A. (1997) Group cohomology, harmonic functions and the first L2-Betti number. Potential Anal., 6(4), 313–326. MR: 98e:20056Google Scholar

Ben Arous, G., Fribergh, A., Gantert, N., and Hammond, A. (2012) Biased random walks on Galton-Watson trees with leaves. Ann. Probab., 40(1), 280–338. MR: 2917774Google Scholar

Ben Arous, G., Fribergh, A., and Sidoravicius, V. (2014) Lyons-Pemantle-Peres monotonicity problem for high biases. Comm. Pure Appl. Math., 67(4), 519–530. MR: 3168120Google Scholar

Ben Arous, G., Hu, Y., Olla, S., and Zeitouni, O. (2013) Einstein relation for biased random walk on Galton-Watson trees. Ann. Inst. Henri Poincaré Probab. Stat., 49(3), 698–721. MR: 3112431Google Scholar

Benassi, A. (1996) Arbres et grandes déviations. In Chauvin, B., Cohen, S., and Rouault, A., editors, Trees, vol. 40 of Progr. Probab., pages 135–140. Birkhäuser, Basel. Proceedings of the Workshop held in Versailles, June 14–16, 1995. MR: 1439977

Benedetti, R. and Petronio, C. (1992) Lectures on Hyperbolic Geometry. Universitext. Springer-Verlag, Berlin. MR: 94e:57015

Benjamini, I. (1991) Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theoret. Probab., 4(3), 631–637. MR: 1115166Google Scholar

Benjamini, I. and Curien, N. (2012) Ergodic theory on stationary random graphs. Electron. J. Probab., 17, paper no. 93, 20 pp. MR: 2994841Google Scholar

Benjamini, I., Duminil-Copin, H., Kozma, G., and Yadin, A. (2015a) Disorder, entropy and harmonic functions. Ann. Probab., 43(5), 2332–2373.Google Scholar

Benjamini, I., Duminil-Copin, H., Kozma, G., and Yadin, A. (2015b) Minimal harmonic functions I, upper bounds. In preparation.

Benjamini, I., Gurel-Gurevich, O., and Lyons, R. (2007) Recurrence of random walk traces. Ann. Probab., 35(2), 732–738. MR: 2308594Google Scholar

Benjamini, I., Kesten, H., Peres, Y., and Schramm, O. (2004) Geometry of the uniform spanning forest: Transitions in dimensions 4; 8; 12;. Ann. of Math. (2), 160(2), 465–491. MR: 2123930Google Scholar

Benjamini, I. and Kozma, G. (2005) A resistance bound via an isoperimetric inequality. Combinatorica, 25(6), 645–650. MR: 2199429Google Scholar

Benjamini, I., Kozma, G., and Schapira, B. (2011) A balanced excited random walk. C. R. Math. Acad. Sci. Paris, 349(7–8), 459–462. MR: 2788390Google Scholar

Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (1999a) Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab., 27(3), 1347–1356. MR: 1733 151Google Scholar

Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (1999b) Group-invariant percolation on graphs. Geom. Funct. Anal., 9(1), 29–66. MR: 99m:60149Google Scholar

Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (2001) Uniform spanning forests. Ann. Probab., 29(1), 1–65. MR: 1825 141Google Scholar

Benjamini, I., Lyons, R., and Schramm, O. (1999) Percolation perturbations in potential theory and random walks. In Picardello, M. and Woess, W., editors, Random Walks and Discrete Potential Theory, Sympos. Math., XXXIX, pages 56–84. Cambridge University Press, Cambridge. Proceedings of the conference held in Cortona, June 1997. MR: 1802426

Benjamini, I. and Müller, S. (2012) On the trace of branching random walks. Groups Geom. Dyn., 6(2), 231–247. MR: 2914859Google Scholar

Benjamini, I., Nachmias, A., and Peres, Y. (2011) Is the critical percolation probability local? Probab. Theory Related Fields, 149(1–2), 261–269. MR: 2773031Google Scholar

Benjamini, I., Pemantle, R., and Peres, Y. (1995) Martin capacity for Markov chains. Ann. Probab., 23(3), 1332–1346. MR: 96g:60098Google Scholar

Benjamini, I., Pemantle, R., and Peres, Y. (1996) Random walks in varying dimensions. J. Theoret. Probab., 9(1), 231–244. MR: 97a:60092Google Scholar

Benjamini, I., Pemantle, R., and Peres, Y. (1998) Unpredictable paths and percolation. Ann. Probab., 26(3), 1198–1211. MR: 99g:60183Google Scholar

Benjamini, I. and Peres, Y. (1992) Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist., 28(4), 557–592. MR: 94f:60089Google Scholar

Benjamini, I. and Peres, Y. (1994a) Markov chains indexed by trees. Ann. Probab., 22(1), 219–243. MR: 1258875Google Scholar

Benjamini, I. and Peres, Y. (1994b) Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields, 98(1), 91–112. MR: 94m:60141Google Scholar

Benjamini, I. and Schramm, O. (1996a) Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math., 126(3), 565–587. MR: 97k:31009Google Scholar

Benjamini, I. and Schramm, O. (1996b) Percolation beyond Zd, many questions and a few answers. Electron. Comm. Probab., 1, paper no. 8, 71–82 (electronic). MR: 97j:60179Google Scholar

Benjamini, I. and Schramm, O. (1996c) Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab., 24(3), 1219–1238. MR: 98d:60134Google Scholar

Benjamini, I. and Schramm, O. (1997) Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal., 7(3), 403–419. MR: 99b:05032Google Scholar

Benjamini, I. and Schramm, O. (2001a) Percolation in the hyperbolic plane. J. Amer. Math. Soc., 14(2), 487–507. MR: 1815220Google Scholar

Benjamini, I. and Schramm, O. (2001b) Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6, paper no. 23, 13 pp. (electronic). MR: 1873300Google Scholar

Benjamini, I. and Schramm, O. (2004) Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality. In Milman, V.D. and Schechtman, G., editors, Geometric Aspects of Functional Analysis, vol. 1850 of Lecture Notes in Math., pages 73–76. Springer, Berlin. Papers from the Israel Seminar (GAFA) held 2002–2003. MR: 2087152

Bennies, J. and Kersting, G. (2000) A random walk approach to Galton-Watson trees. J. Theoret. Probab., 13(3), 777–803. MR: 1785529Google Scholar

Berestycki, N., Lubetzky, E., Peres, Y., and Sly, A. (2015) Random walks on the random graph. Preprint, http://www.arxiv.org/abs/1504.01999.

Berger, N., Gantert, N., and Peres, Y. (2003) The speed of biased random walk on percolation clusters. Probab. Theory Related Fields, 126(2), 221–242. MR: 1990055Google Scholar

van den Berg, J. and Keane, M. (1984) On the continuity of the percolation probability function. In Beals, R., Beck, A., Bellow, A., and Hajian, A., editors, Conference in Modern Analysis and Probability (New Haven, Conn., 1982), pages 61–65. Amer. Math. Soc., Providence, RI. MR: 85g:60100

van den Berg, J. and Kesten, H. (1985) Inequalities with applications to percolation and reliability. J. Appl. Probab., 22(3), 556–569. MR: 87b:60027Google Scholar

van den Berg, J. and Kesten, H. (1991) Stability properties of a flow process in graphs. Random Structures Algorithms, 2(3), 335–341. MR: 92d:90027Google Scholar

Biggins, J.D (1977) Chernoff's theorem in the branching random walk. J. Appl. Probability, 14(3), 630–636. MR: 0464415Google Scholar

Biggs, N.L, Mohar, B., and Shawe-Taylor, J. (1988) The spectral radius of infinite graphs. Bull. London Math. Soc., 20(2), 116–120. MR: 89a:05103Google Scholar

Billingsley, P. (1965) Ergodic Theory and Information. John Wiley, New York. MR: 33:254

Billingsley, P. (1995) Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. John Wiley, New York. MR: 1324786

Björklund, M. (2014) Five remarks about random walks on groups. Preprint, http://www.arxiv.org/abs/1406.0763.

Blachère, S., Haïssinsky, P., and Mathieu, P. (2008) Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab., 36(3), 1134–1152. MR: 2408585Google Scholar

Blackwell, D. (1955) On transient Markov processes with a countable number of states and stationary transition probabilities. Ann. Math. Statist., 26, 654–658. MR: 17,754dGoogle Scholar

Boley, D., Ranjan, G., and Zhang, Z.L. (2011) Commute times for a directed graph using an asymmetric Laplacian. Linear Algebra Appl., 435(2), 224–242. MR: 2782776Google Scholar

Bollobás, B. (1998) Modern Graph Theory. Vol. 184 of Graduate Texts in Mathematics. Springer-Verlag, New York. MR: 99h:05001

Borcea, J., Brändén, P., and Liggett, T.M. (2009) Negative dependence and the geometry of polynomials. J. Amer. Math. Soc., 22(2), 521–567. MR: 2476782Google Scholar

Borre, K. and Meissl, P. (1974) Strength analysis of leveling-type networks. An application of random walk theory. Geodaet. Inst. Medd., 50, 80. MR: 0475698Google Scholar

Borwein, J.M and Zucker, I.J. (1992) Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal., 12(4), 519–526. MR: 1186733Google Scholar

Bourgain, J. (1985) On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math., 52(1–2), 46–52. MR: 815600Google Scholar

Bowen, L. (2004) Couplings of uniform spanning forests. Proc. Amer. Math. Soc., 132(7), 2151–2158 (electronic). MR: 2053 989Google Scholar

Brieussel, J. (2009) Amenability and non-uniform growth of some directed automorphism groups of a rooted tree. Math. Z., 263(2), 265–293. MR: 2534118Google Scholar

Brieussel, J. (2013) Behaviors of entropy on finitely generated groups. Ann. Probab., 41(6), 4116–4161. MR: 3161471Google Scholar

Brieussel, J. and Zheng, T. (2015) Speed of random walks, isoperimetry and compression of finitely generated groups. Preprint, http://www.arxiv.org/abs/1510.08040.

Broadbent, S.R and Hammersley, J.M. (1957) Percolation processes. I. Crystals and mazes. Proc. Cambridge Philos. Soc., 53, 629–641. MR: 0091567Google Scholar

Broder, A. (1989) Generating random spanning trees. In 30th Annual Symposium on Foundations of Computer Science (Research Triangle Park, North Carolina), pages 442–447. IEEE, New York. http://dx.doi.org/10.1109/SFCS.1989.63516.

Broder, A.Z and Karlin, A.R. (1989) Bounds on the cover time. J. Theoret. Probab., 2(1), 101–120. MR: 981768Google Scholar

Brofferio, S. and Schapira, B. (2011) Poisson boundary of GLd. Israel J. Math., 185, 125–140. MR: 2837130Google Scholar

Broman, E.I, Camia, F., Joosten, M., and Meester, R. (2013) Dimension (in)equalities and Hölder continuous curves in fractal percolation. J. Theoret. Probab., 26(3), 836–854. MR: 3090553Google Scholar

Brooks, R.L, Smith, C.A.B., Stone, A.H., and Tutte, W.T. (1940) The dissection of rectangles into squares. Duke Math. J., 7, 312–340. MR: 2,153dGoogle Scholar

Burton, R.M and Keane, M. (1989) Density and uniqueness in percolation. Comm. Math. Phys., 121(3), 501–505. MR: 90g:60090Google Scholar

Burton, R.M and Pemantle, R. (1993) Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer- impedances. Ann. Probab., 21(3), 1329–1371. MR: 94m:60019Google Scholar

Buser, P. (1982) A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4), 15(2), 213–230. MR: 683635Google Scholar

Campanino, M. (1985) Inequalities for critical probabilities in percolation. In Durrett, R., editor, Particle Systems, Random Media and Large Deviations, vol. 41 of Contemp. Math., pages 1–9. Amer. Math. Soc., Providence, RI. Proceedings of the AMS-IMS-SIAM joint summer research conference in the mathematical sciences on mathematics of phase transitions held at Bowdoin College, Brunswick, Maine, June 24–30, 1984. MR: 814699

Campbell, G.A (1911) Cisoidal oscillations. Trans. Amer. Inst. Electrical Engineers, 30, 873–909. http://dx.doi.org/10.1109/T-AIEE.1911.4768303.Google Scholar

Cannon, J.W, Floyd, W.J., Kenyon, R.W., and Parry, W.R. (1997) Hyperbolic geometry. In Levy, S., editor, Flavors of Geometry, vol. 31 of Mathematical Sciences Research Institute Publications, pages 59–115. Cambridge University Press, Cambridge. MR: 1491098

Cannon, J.W, Floyd, W.J., and Parry, W.R. (1994) Squaring rectangles: The finite Riemann mapping theorem. In Abikoff, W., Birman, J.S., and Kuiken, K., editors, The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions, vol. 169 of Contemp. Math., pages 133–212. Amer. Math. Soc., Providence, RI. Proceedings of the conference held at Polytechnic University, Brooklyn, New York, May 1–3, 1992. MR: 1292901

Caputo, P., Liggett, T.M., and Richthammer, T. (2010) Proof of Aldous' spectral gap conjecture. J. Amer. Math. Soc., 23(3), 831–851. MR: 2629990Google Scholar

Carleson, L. (1967) Selected Problems on Exceptional Sets. Vol. 13 of Van Nostrand Mathematical Studies. D. Van Nostrand, Princeton, NJ. MR: 0225986

Carlitz, L., Wilansky, A., Milnor, J., Struble, R.A., Felsinger, N., Simoes, J.M.S., Power, E.A., Shafer, R.E., and Maas, R.E. (1968) Problems and Solutions: Advanced Problems: 5600–5609. Amer. Math. Monthly, 75(6), 685–687. MR: 1534960Google Scholar

Carmesin, J. (2012) A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy. Potential Anal., 37(3), 229–245. MR: 2969301Google Scholar

Carmesin, J. and Georgakopoulos, A. (2015) Every planar graph with the Liouville property is amenable. Preprint, http://www.arxiv.org/abs/1502.02542.

Carne, T.K (1985) A transmutation formula for Markov chains. Bull. Sci. Math. (2), 109(4), 399–405. MR: 87m:60142Google Scholar

Cartwright, D.I, Kaimanovich, V.A., and Woess, W. (1994) Random walks on the affine group of local fields and of homogeneous trees. Ann. Inst. Fourier (Grenoble), 44(4), 1243–1288. MR: 1306556Google Scholar

Cartwright, D.I and Soardi, P.M. (1989) Convergence to ends for random walks on the automorphism group of a tree. Proc. Amer. Math. Soc., 107(3), 817–823. MR: 90f:60137Google Scholar

Cayley, A. (1889) A theorem on trees. Quart. J. Math., 23, 376–378. http://dx.doi.org/10.1017/CBO9780511703799.010.Google Scholar

Teixeira, A.Q. (2012) From Random Walk Trajectories to Random Interlacements. Vol. 23 of Ensaios Matemáticos [Mathe- matical Surveys]. Sociedade Brasileira de Matemática, Rio de Janeiro. MR: 3014964

Chaboud, T. and Kenyon, C. (1996) Planar Cayley graphs with regular dual. Internat. J. Algebra Comput., 6(5), 553–561. MR: 98a:05077Google Scholar

Chalupa, J., Reich, G.R., and Leath, P.L. (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.Google Scholar

Chandra, A.K, Raghavan, P., Ruzzo, W.L., Smolensky, R., and Tiwari, P. (1996/1997) The electrical resistance of a graph captures its commute and cover times. Comput. Complexity, 6(4), 312–340. MR: 99h:60140Google Scholar

Chauvin, B., Rouault, A., and Wakolbinger, A. (1991) Growing conditioned trees. Stochastic Process. Appl., 39(1), 117–130. MR: 1135089Google Scholar

Chayes, J.T, Chayes, L., and Durrett, R. (1988) Connectivity properties of Mandelbrot's percolation process. Probab. Theory Related Fields, 77(3), 307–324. MR: 931500Google Scholar

Chayes, J.T, Chayes, L., and Newman, C.M. (1985) The stochastic geometry of invasion percolation. Comm. Math. Phys., 101(3), 383–407. MR: 87i:82072Google Scholar

Chayes, L. (1995a) Aspects of the fractal percolation process. In Bandt, C., Graf, S., and Zähle, M., editors, Fractal Geometry and Stochastics, vol. 37 of Progr. Probab., pages 113–143. Birkhäuser, Basel. Papers from the conference held in Finsterbergen, June 12–18, 1994. MR: 1391973

Chayes, L. (1995b) On the absence of directed fractal percolation. J. Phys. A.: Math. Gen., 28, L295–L301. http://dx.doi.org/10.1088/0305-4470/28/10/003.Google Scholar

Chayes, L., Pemantle, R., and Peres, Y. (1997) No directed fractal percolation in zero area. J. Statist. Phys., 88(5–6), 1353–1362. MR: 1478072Google Scholar

Cheeger, J. (1970) A lower bound for the smallest eigenvalue of the Laplacian. In Gunning, R.C., editor, Problems in Analysis, pages 195–199. Princeton University Press, Princeton, NJ. A symposium in honor of Salomon Bochner, Princeton University, Princeton, NJ, 1–3 April 1969. MR: 53:6645

Cheeger, J. and Gromov, M. (1986) L2-cohomology and group cohomology. Topology, 25(2), 189–215. MR: 87i:58161Google Scholar

Chen, D. (1997) Average properties of random walks on Galton-Watson trees. Ann. Inst. H. Poincaré Probab. Statist., 33(3), 359–369. MR: 1457056Google Scholar

Chen, D. and Peres, Y. (2004) Anchored expansion, percolation and speed. Ann. Probab., 32(4), 2978–2995. With an appendix by Gábor Pete. MR: 2094436Google Scholar

Chen, D. and Zhang, F.X. (2007) On the monotonicity of the speed of random walks on a percolation cluster of trees. Acta Math. Sin. (Engl. Ser.), 23(11), 1949–1954. MR: 2359112Google Scholar

Choquet, G. and Deny, J. (1960) Sur l'équation de convolution. C. R. Acad. Sci. Paris, 250, 799–801. MR: 22:9808Google Scholar

Chung, F.RK., Graham, R.L., Frankl, P., and Shearer, J.B. (1986) Some intersection theorems for ordered sets and graphs. J. Combin. Theory Ser. A, 43(1), 23–37. MR: 859293Google Scholar

Chung, F.RK.and Tetali, P. (1998) Isoperimetric inequalities for Cartesian products of graphs. Combin. Probab. Comput., 7(2), 141–148. MR: 2000c:05085Google Scholar

Colbourn, C.J, Provan, J.S., and Vertigan, D. (1995) A new approach to solving three combinatorial enumeration problems on planar graphs. Discrete Appl. Math., 60(1–3), 119–129. MR: 96e:05154Google Scholar

Collevecchio, A. (2006) On the transience of processes defined on Galton-Watson trees. Ann. Probab., 34(3), 870–878. MR: 2243872Google Scholar

Constantine, G.M (2003) Graphs, networks, and linear unbiased estimates. Discrete Appl. Math., 130(3), 381–393. MR: 1999697Google Scholar

Coppersmith, D., Doyle, P., Raghavan, P., and Snir, M. (1993) Random walks on weighted graphs and applications to on-line algorithms. J. Assoc. Comput. Mach., 40(3), 421–453. MR: 1370357Google Scholar

Coppersmith, D., Feige, U., and Shearer, J. (1996) Random walks on regular and irregular graphs. SIAM J. Discrete Math., 9(2), 301–308. MR: 1386885Google Scholar

Coppersmith, D., Tetali, P., and Winkler, P. (1993) Collisions among random walks on a graph. SIAM J. Discrete Math., 6(3), 363–374. MR: 1229691Google Scholar

Coulhon, T. (1996) Ultracontractivity and Nash type inequalities. J. Funct. Anal., 141(2), 510–539. MR: 1418518Google Scholar

Coulhon, T., Grigor'yan, A., and Pittet, C. (2001) A geometric approach to on-diagonal heat kernel lower bounds on groups. Ann. Inst. Fourier (Grenoble), 51(6), 1763–1827. MR: 1871289Google Scholar

Coulhon, T. and Saloff-Coste, L. (1993) Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana, 9(2), 293–314. MR: 94g:58263Google Scholar

Cournot, A.A (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.

Cox, J.T and Durrett, R. (1983) Oriented percolation in dimensions: Bounds and asymptotic formulas. Math. Proc. Cambridge Philos. Soc., 93(1), 151–162. MR: 84e:60150Google Scholar

Croydon, D. (2008) Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. Ann. Inst. Henri Poincaré Probab. Stat., 44(6), 987–1019. MR: 2469332Google Scholar

Croydon, D. and Kumagai, T. (2008) Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive. Electron. J. Probab., 13, paper no. 51, 1419–1441. MR: 2438812Google Scholar

Curien, N. and Le Gall, J.F. (2011) Random recursive triangulations of the disk via fragmentation theory. Ann. Probab., 39(6), 2224–2270. MR: 2932668Google Scholar

Curien, N. and Le Gall, J.F. (2016) The harmonic measure of balls in random trees. Ann. Probab. To appear, http://www.arxiv.org/abs/1304.7190.

Curien, N. and Peres, Y. (2011) Random laminations and multitype branching processes. Electron. Commun. Probab., 16, 435–446. MR: 2831082Google Scholar

Dai, J.J (2005) A once edge-reinforced random walk on a Galton-Watson tree is transient. Statist. Probab. Lett., 73(2), 115–124. MR: 2159246Google Scholar

Davenport, H., Erdʺos, P., and LeVeque, W.J. (1963) On Weyl's criterion for uniform distribution. Michigan Math. J., 10, 311–314. MR: 0153656Google Scholar

Dekking, F.M and Meester, R.W.J. (1990) On the structure of Mandelbrot's percolation process and other random Cantor sets. J. Statist. Phys., 58(5–6), 1109–1126. MR: 1049059Google Scholar

Dellacherie, C. and Meyer, P.A. (1978) Probabilities and Potential. Vol. 29 of North-Holland Mathematics Studies. North-Holland, Amsterdam. MR: 521810

Dembo, A. (2005) Favorite points, cover times and fractals. In Lectures on Probability Theory and Statistics, vol. 1869 of Lecture Notes in Math., pages 1–101. Springer, Berlin. Lectures from the 33rd Probability Summer School held in Saint-Flour, July 6–23, 2003, edited by Jean Picard. MR: 2228383

Dembo, A., Gantert, N., Peres, Y., and Zeitouni, O. (2002) Large deviations for random walks on Galton-Watson trees: Averaging and uncertainty. Probab. Theory Related Fields, 122(2), 241–288. MR: 1894069Google Scholar

Dembo, A., Gantert, N., and Zeitouni, O. (2004) Large deviations for random walk in random environment with holding times. Ann. Probab., 32(1B), 996–1029. MR: 2044672Google Scholar

Dembo, A., Peres, Y., Rosen, J., and Zeitouni, O. (2001) Thick points for planar Brownian motion and the Erdʺos-Taylor conjecture on random walk. Acta Math., 186(2), 239–270. MR: 1846031Google Scholar

Dembo, A. and Zeitouni, O. (1998) Large Deviations Techniques and Applications, 2nd ed. Vol. 38 of Applications of Mathematics. Springer-Verlag, New York. MR: 1619036

Derriennic, Y. (1976) Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires. Ann. Inst. H. Poincaré Sect. B (N.S.), 12(2), 111–129. MR: 0423532Google Scholar

Derriennic, Y. (1980) Quelques applications du théorème ergodique sous-additif. In Journées sur les Marches Aléatoires, vol. 74 of Astérisque, pages 183–201, 4. Soc. Math. France, Paris. Held at Kleebach, March 5–10, 1979. MR: 588163

Deza, M.M and Laurent, M. (1997) Geometry of Cuts and Metrics. Vol. 15 of Algorithms and Combinatorics. Springer-Verlag, Berlin. MR: 1460488

Diaconis, P. and Evans, S.N. (2002) A different construction of Gaussian fields from Markov chains: Dirichlet covariances. Ann. Inst. H. Poincaré Probab. Statist., 38(6), 863–878. MR: 1955341Google Scholar

Diaconis, P. and Fill, J.A. (1990) Strong stationary times via a new form of duality. Ann. Probab., 18(4), 1483–1522. MR: 1071805Google Scholar

Diaconis, P. and Saloff-Coste, L. (1994) Moderate growth and random walk on finite groups. Geom. Funct. Anal., 4(1), 1–36. MR: 1254308Google Scholar

Diestel, R. and Leader, I. (2001) A conjecture concerning a limit of non-Cayley graphs. J. Algebraic Combin., 14(1), 17–25. MR: 2002h:05082Google Scholar

Ding, J., Lee, J.R., and Peres, Y. (2012) Cover times, blanket times, and majorizing measures. Ann. of Math. (2), 175(3), 1409–1471. MR: 2912708Google Scholar

Ding, J., Lee, J.R., and Peres, Y. (2013) Markov type and threshold embeddings. Geom. Funct. Anal., 23(4), 1207–1229. MR: 3077911Google Scholar

Disertori, M., Sabot, C., and Tarrès, P. (2015) Transience of edge-reinforced random walk. Comm. Math. Phys., 339(1), 121–148. MR: 3366053Google Scholar

Dodziuk, J. (1977) de Rham-Hodge theory for L2-cohomology of infinite coverings. Topology, 16(2), 157–165. MR: 0445560Google Scholar

Dodziuk, J. (1979) L2 harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Amer. Math. Soc., 77(3), 395–400. MR: 81e:58004Google Scholar

Dodziuk, J. (1984) Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc., 284(2), 787–794. MR: 85m:58185Google Scholar

Dodziuk, J. and Kendall, W.S. (1986) Combinatorial Laplacians and isoperimetric inequality. In Elworthy, K.D., editor, From Local Times to Global Geometry, Control and Physics, pages 68–74. Longman Sci. Tech., Harlow. Papers from the Warwick symposium on stochastic differential equations and applications, held at the University of Warwick, Coventry, 1984/85. MR: 88h:58118

Doob, J.L (1984) Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, New York. MR: 85k:31001

Doyle, P.G (1988) Electric currents in infinite networks. Unpublished, http://www.arxiv.org/abs/math/0703899.

Doyle, P.G (2009) The Kemeny constant of a Markov chain. Unpublished, http://www.arxiv.org/abs/0909.2636.

Doyle, P.G and Snell, J.L. (1984) Random Walks and Electric Networks. Mathematical Association of America, Washington, DC. Also available at http://arxiv.org/abs/math/0001057. MR: 89a:94023

Doyle, P.G and Steiner, J. (2008) Commuting time geometry of ergodic Markov chains. Unpublished, http://www.arxiv.org/abs/1107.2612.

Drewitz, A., Ráth, B., and Sapozhnikov, A. (2014) An Introduction to Random Interlacements. Springer Briefs in Mathematics. Springer, Cham. MR: 3308116

Dubins, L.E and Freedman, D.A. (1967) Random distribution functions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), pages Vol. II: Contributions to Probability Theory, Part 1, pp. 183–214. University of California Press, Berkeley. MR: 0214109

Duffin, R.J (1962) The extremal length of a network. J. Math. Anal. Appl., 5, 200–215. MR: 0143468Google Scholar

Duminil-Copin, H., Sidoravicius, V., and Tassion, V. (2016) Absence of infinite cluster for critical Bernoulli percolation on slabs. Comm. Pure Appl. Math., 69(7), 1397–1411. MR: 3503025Google Scholar

Duminil-Copin, H. and Smirnov, S. (2012) The connective constant of the honeycomb lattice equals. Ann. of Math. (2), 175(3), 1653–1665. MR: 2912714Google Scholar

Duminil-Copin, H. and Tassion, V. (2015) A new proof of the sharpness of the phase transition for Bernoulli percolation on.d. Enseign. Math. To appear, http://www.arxiv.org/abs/1502.03051.

Duminil-Copin, H. and Tassion, V. (2016) A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Comm. Math. Phys., 343(2), 725–745. MR: 3477351Google Scholar

Duplantier, B. (1992) Loop-erased self-avoiding walks in 2D. Physica A, 191, 516–522. http://dx.doi.org/10.1016/ 0378-4371(92)90575-B.Google Scholar

Duquesne, T. and Le Gall, J.F. (2002) Random trees, Lévy processes and spatial branching processes. Astérisque, 281, vi+147. MR: 1954248Google Scholar

Durrett, R. (1986) Reversible diffusion processes. In Chao, J.A. and Woyczyński, W.A., editors, Probability Theory and Harmonic Analysis, pages 67–89. Dekker, New York. Papers from the conference held in Cleveland, Ohio, May 10–12, 1983. MR: 88b:60175

Durrett, R. (2010) Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. MR: 2722836

Durrett, R., Kesten, H., and Lawler, G. (1991) Making money from fair games. In Durrett, R. and Kesten, H., editors, Random Walks, Brownian Motion, and Interacting Particle Systems, vol. 28 of Progr. Probab., pages 255–267. Birkhäuser, Boston. A Festschrift in honor of Frank Spitzer. MR: 1146451

Dvoretzky, A. (1949) On the strong stability of a sequence of events. Ann. Math. Statistics, 20, 296–299. MR: 0031675Google Scholar

Dvoretzky, A. and Erdös, P. (1951) Some problems on random walk in space. In Proc. Second Berkeley Symposium on Math. Statist. and Probability, 1950, pages 353–367. University of California Press, Berkeley. MR: 0047272

Dvoretzky, A., Erdös, P., and Kakutani, S. (1950) Double points of paths of Brownian motion in n-space. Acta Sci. Math. Szeged, 12(Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B), 75–81. MR: 0034972Google Scholar

Dvoretzky, A., Erdʺos, P., Kakutani, S., and Taylor, S.J. (1957) Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc., 53, 856–862. MR: 0094855Google Scholar

Dwass, M. (1969) The total progeny in a branching process and a related random walk. J. Appl. Probability, 6, 682–686. MR: 0253433Google Scholar

Dymarz, T. (2010) Bilipschitz equivalence is not equivalent to quasi-isometric equivalence for finitely generated groups. Duke Math. J., 154(3), 509–526. MR: 2730576Google Scholar

Dynkin, E.B (1969) The boundary theory of Markov processes (discrete case). Uspehi Mat. Nauk, 24(2), 3–42. English translation: Russ. Math. Surv. 24 (1969), no. 2, 1–42; http://dx.doi.org/10.1070/RM1969v024n02ABEH001341. MR: 0245096Google Scholar

Dynkin, E.B (1980) Markov processes and random fields. Bull. Amer. Math. Soc. (N.S.), 3(3), 975–999. MR: 585179Google Scholar

Dynkin, E.B and Maljutov, M.B. (1961) Random walk on groups with a finite number of generators. Dokl. Akad. Nauk SSSR, 137, 1042–1045. English translation: Soviet Math. Dokl. (1961) 2, 399–402. MR: 24:A1751Google Scholar

Eckmann, B. (2000) Introduction to l2-methods in topology: Reduced l2-homology, harmonic chains, l2-Betti numbers. Israel J. Math., 117, 183–219. Notes prepared by Guido Mislin. MR: 1760592Google Scholar

Edgar, G.A (1990) Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. Springer-Verlag, New York. MR: 1065392

Edmonds, J. (1971) Matroids and the greedy algorithm. Math. Programming, 1, 127–136. MR: 0297357Google Scholar

Enflo, P. (1969) On the nonexistence of uniform homeomorphisms between Lp-spaces. Ark. Mat., 8, 103–105 (1969). MR: 0271719Google Scholar

Epstein, I. and Hjorth, G. (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.

Erschler, A. (2010) Poisson-Furstenberg boundaries, large-scale geometry and growth of groups. In Proceedings of the International Congress of Mathematicians. Volume II, pages 681–704. Hindustan Book Agency, New Delhi. MR: 2827814

Erschler, A. (2011) Poisson-Furstenberg boundary of random walks on wreath products and free metabelian groups. Comment. Math. Helv., 86(1), 113–143. MR: 2745278Google Scholar

Erschler, A. and Karlsson, A. (2010) Homomorphisms to. constructed from random walks. Ann. Inst. Fourier (Grenoble), 60(6), 2095–2113. MR: 2791651Google Scholar

Eskin, A., Fisher, D., and Whyte, K. (2012) Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs. Ann. of Math. (2), 176(1), 221–260. MR: 2925383Google Scholar

Ethier, S.N and Lee, J. (2009) Limit theorems for Parrondo's paradox. Electron. J. Probab., 14, paper no. 62, 1827–1862. MR: 2540850Google Scholar

Evans, S.N (1987) Multiple points in the sample paths of a Lévy process. Probab. Theory Related Fields, 76(3), 359–367. MR: 912660Google Scholar

Falconer, K.J (1986) Random fractals. Math. Proc. Cambridge Philos. Soc., 100(3), 559–582. MR: 88e:28005Google Scholar

Falconer, K.J (1987) Cut-set sums and tree processes. Proc. Amer. Math. Soc., 101(2), 337–346. MR: 88m:90052Google Scholar

Falconer, K.J (1990) Fractal Geometry. Mathematical foundations and applications. John Wiley, Chichester. MR: 1102677

Fan, A.H (1989) Décompositions de Mesures et Recouvrements Aléatoires. Ph.D. thesis, Université de Paris-Sud, Département de Mathématique, Orsay. MR: 91e:60009

Fan, A.H (1990) Sur quelques processus de naissance et de mort. C. R. Acad. Sci. Paris Sér. I Math., 310(6), 441–444. MR: 91d:60103Google Scholar

Faraud, G. (2011) A central limit theorem for random walk in a random environment on marked Galton-Watson trees. Electron. J. Probab., 16, paper no. 6, 174–215. MR: 2754802Google Scholar

Faraud, G., Hu, Y., and Shi, Z. (2012) Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Related Fields, 154(3–4), 621–660. MR: 3000557Google Scholar

Feder, T. and Mihail, M. (1992) Balanced matroids. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pages 26–38. Association for Computing Machinery (ACM), New York. http://dx.doi.org/10.1145/129712.129716.

Feller, W. (1971) An Introduction to Probability Theory and Its Applications. Vol. II., 2nd ed. John Wiley, New York. MR: 42:5292

Fisher, M.E (1961) Critical probabilities for cluster size and percolation problems. J. Math. Phys., 2, 620–627. MR: 0126306Google Scholar

Fitzner, R. and van der Hofstad, R. (2015) Nearest-neighbor percolation function is continuous for d > 10. Preprint, http://www.arxiv.org/abs/1506.07977.

Fitzsimmons, P.J and Salisbury, T.S. (1989) Capacity and energy for multiparameter Markov processes. Ann. Inst. H. Poincaré Probab. Statist., 25(3), 325–350. MR: 1023955Google Scholar

Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics. Cambridge University Press, Cambridge. MR: 2483235

Flanders, H. (1971) Infinite networks. I: Resistive networks. IEEE Trans. Circuit Theory, CT–18, 326–331. MR: 0275998Google Scholar

Flanders, H. (1974) A new proof of R. Foster's averaging formula in networks. Linear Algebra and Appl., 8, 35–37. MR: 0329772Google Scholar

Fleischmann, K. and Siegmund-Schultze, R. (1977) The structure of reduced critical Galton-Watson processes. Math. Nachr., 79, 233–241. MR: 0461689Google Scholar

Foguel, S.R (1971) On the “zero-two” law. Israel J. Math., 10, 275–280. MR: 0298759Google Scholar

Foguel, S.R (1976) More on the “zero-two” law. Proc. Amer. Math. Soc., 61(2), 262–264 (1977). MR: 0428076Google Scholar

Følner, E. (1955) On groups with full Banach mean value. Math. Scand., 3, 243–254. MR: 18,51fGoogle Scholar

Fontes, L.R, Schonmann, R.H., and Sidoravicius, V. (2002) Stretched exponential fixation in stochastic Ising models at zero temperature. Comm. Math. Phys., 228(3), 495–518. MR: 1918786Google Scholar

Ford, L.RJr. and Fulkerson, D.R. (1962) Flows in Networks. Princeton University Press, Princeton, NJ. MR: 28:2917

Fortuin, C.M (1972a) On the random-cluster model. II. The percolation model. Physica, 58, 393–418. MR: 51:14826Google Scholar

Fortuin, C.M (1972b) On the random-cluster model. III. The simple random-cluster model. Physica, 59, 545–570. MR: 55:5127Google Scholar

Fortuin, C.M and Kasteleyn, P.W. (1972) On the random-cluster model. I. Introduction and relation to other models. Physica, 57, 536–564. MR: 0359655Google Scholar

Fortuin, C.M, Kasteleyn, P.W., and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Comm. Math. Phys., 22, 89–103. MR: 46:8607Google Scholar

Foster, R.M (1948) The average impedance of an electrical network. In Reissner Anniversary Volume, Contributions to Applied Mechanics, pages 333–340. J. W. Edwards, Ann Arbor, Michigan. Edited by the Staff of the Department of Aeronautical Engineering and Applied Mechanics of the Polytechnic Institute of Brooklyn. MR: 10,662a

Frostman, O. (1935) Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddel. Lunds Univ. Mat. Sem., 3, 1–118.Google Scholar

Fukushima, M. (1980) Dirichlet Forms and Markov Processes. North-Holland, Amsterdam. MR: 81f:60105

Fukushima, M. (1985) Energy forms and diffusion processes. In Streit, L., editor, Mathematics + Physics. Vol. 1, pages 65–97. World Scientific, Singapore. Lectures on recent results. MR: 87m:60176

Furstenberg, H. (1963) A Poisson formula for semi-simple Lie groups. Ann. of Math. (2), 77, 335–386. MR: 0146298Google Scholar

Furstenberg, H. (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1, 1–49. MR: 35:4369Google Scholar

Furstenberg, H. (1970) Intersections of Cantor sets and transversality of semigroups. In Gunning, R.C., editor, Problems in Analysis, pages 41–59. Princeton University Press, Princeton, NJ. A symposium in honor of Salomon Bochner, Princeton University, Princeton, NJ, 1–3 April 1969. MR: 50:7040

Furstenberg, H. (1971a) Boundaries of Lie groups and discrete subgroups. In Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pages 301–306. Gauthier-Villars, Paris. MR: 0430160

Furstenberg, H. (1971b) Random walks and discrete subgroups of Lie groups. In Advances in Probability and Related Topics, Vol. 1, pages 1–63. Dekker, New York. MR: 0284569

Furstenberg, H. (1973) Boundary theory and stochastic processes on homogeneous spaces. In Moore, C.C., editor, Harmonic Analysis on Homogeneous Spaces, pages 193–229. Amer. Math. Soc., Providence R.I. Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972). MR: 0352328

Furstenberg, H. (2008) Ergodic fractal measures and dimension conservation. Ergodic Theory Dynam. Systems, 28(2), 405–422. MR: 2408385Google Scholar

Furstenberg, H. and Weiss, B. (2003) Markov processes and Ramsey theory for trees. Combin. Probab. Comput., 12(5–6), 547–563. MR: 2037069Google Scholar

Gaboriau, D. (1998) Mercuriale de groupes et de relations. C. R. Acad. Sci. Paris Sér. I Math., 326(2), 219–222. MR: 99h:28034Google Scholar

Gaboriau, D. (2000) Coût des relations d'équivalence et des groupes. Invent. Math., 139(1), 41–98. MR: 1728 876Google Scholar

Gaboriau, D. (2002) Invariants l2 de relations d'équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci., 95, 93–150. MR: 1953 191Google Scholar

Gaboriau, D. (2005) Invariant percolation and harmonic Dirichlet functions. Geom. Funct. Anal., 15(5), 1004–1051. MR: 2221157Google Scholar

Gaboriau, D. and Lyons, R. (2009) A measurable-group-theoretic solution to von Neumann's problem. Invent. Math., 177(3), 533–540. MR: 2534099Google Scholar

Gaboriau, D. and Tucker-Drob, R. (2015) Approximations of standard equivalence relations and Bernoulli percolation at pu. Preprint, http://www.arxiv.org/abs/1509.00247.

Gandolfi, A., Keane, M.S., and Newman, C.M. (1992) Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields, 92(4), 511–527. MR: 93f:60149Google Scholar

Gantert, N., Müller, S., Popov, S., and Vachkovskaia, M. (2012) Random walks on Galton-Watson trees with random conductances. Stochastic Process. Appl., 122(4), 1652–1671. MR: 2914767Google Scholar

Garban, C., Pete, G., and Schramm, O. (2013) The scaling limits of the minimal spanning tree and invasion percolation in the plane. Preprint, http://www.arxiv.org/abs/1309.0269.

Garnett, J.B and Marshall, D.E. (2005) Harmonic Measure. Vol. 2 of New Mathematical Monographs. Cambridge University Press, Cambridge. MR: 2150803

Gaudillière, A. and Landim, C. (2014) A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Related Fields, 158(1–2), 55–89. MR: 3152780Google Scholar

Gautero, F. and Mathéus, F. (2012) Poisson boundary of groups acting on.-trees. Israel J. Math., 191(2), 585–646. MR: 3011489Google Scholar

Geiger, J. (1995) Contour processes of random trees. In Etheridge, A., editor, Stochastic Partial Differential Equations, vol. 216 of London Math. Soc. Lecture Note Ser., pages 72–96. Cambridge University Press, Cambridge. Papers from the workshop held at the University of Edinburgh, Edinburgh, March 1994. MR: 1352736

Geiger, J. (1999) Elementary new proofs of classical limit theorems for Galton-Watson processes. J. Appl. Probab., 36(2), 301–309. MR: 1724 856Google Scholar

Georgakopoulos, A. (2010) Lamplighter graphs do not admit harmonic functions of finite energy. Proc. Amer. Math. Soc., 138(9), 3057–3061. MR: 2653930Google Scholar

Georgakopoulos, A. (2016) The boundary of a square tiling of a graph coincides with the Poisson boundary. Invent. Math., 203(3), 773–821. MR: 3461366Google Scholar

Georgakopoulos, A. and Winkler, P. (2014) New bounds for edge-cover by random walk. Combin. Probab. Comput., 23(4), 571–584. MR: 3217361Google Scholar

Gerl, P. (1988) Random walks on graphs with a strong isoperimetric property. J. Theoret. Probab., 1(2), 171–187. MR: 89g:60216Google Scholar

Gilch, L.A (2007) Rate of escape of random walks on free products. J. Aust. Math. Soc., 83(1), 31–54. MR: 2378433Google Scholar

Gilch, L.A (2011) Asymptotic entropy of random walks on free products. Electron. J. Probab., 16, paper no. 3, 76–105. MR: 2749773Google Scholar

Glasner, E. and Weiss, B. (1997) Kazhdan's property T and the geometry of the collection of invariant measures. Geom. Funct. Anal., 7(5), 917–935. MR: 99f:28029Google Scholar

Glasser, M.L and Zucker, I.J. (1977) Extended Watson integrals for the cubic lattices. Proc. Nat. Acad. Sci. U.S.A., 74(5), 1800–1801. MR: 0442300Google Scholar

Gneiting, T. and Raftery, A.E. (2007) Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc., 102(477), 359–378. MR: 2345548Google Scholar

Godsil, C.D, Imrich, W., Seifter, N., Watkins, M.E., and Woess, W. (1989) A note on bounded automorphisms of infinite graphs. Graphs Combin., 5(4), 333–338. MR: 1032384Google Scholar

Goldman, J.R and Kauffman, L.H. (1993) Knots, tangles, and electrical networks. Adv. in Appl. Math., 14(3), 267–306. MR: 94m:57013Google Scholar

Gouëzel, S., Mathéus, F., and Maucourant, F. (2015) Sharp lower bounds for the asymptotic entropy of symmetric random walks. Groups Geom. Dyn., 9(3), 711–735. MR: 3420541Google Scholar

Grabowski, L. (2014) On Turing dynamical systems and the Atiyah problem. Invent. Math., 198(1), 27–69. MR: 3260857Google Scholar

Graf, S. (1987) Statistically self-similar fractals. Probab. Theory Related Fields, 74(3), 357–392. MR: 88c:60038Google Scholar

Graf, S., Mauldin, R.D., and Williams, S.C. (1988) The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc., 71(381), x+121. MR: 88k:28010Google Scholar

Gravner, J. and Holroyd, A.E. (2009) Local bootstrap percolation. Electron. J. Probab., 14, paper no. 14, 385–399. MR: 2480546Google Scholar

Grey, D.R (1980) A new look at convergence of branching processes. Ann. Probab., 8(2), 377–380. MR: 81e:60091Google Scholar

Griffeath, D. and Liggett, T.M. (1982) Critical phenomena for Spitzer's reversible nearest particle systems. Ann. Probab., 10(4), 881–895. MR: 84f:60140Google Scholar

Grigorchuk, R.I (1980) Symmetrical random walks on discrete groups. In Dobrushin, R.L., Sinai, Ya.G., and Griffeath, D., editors, Multicomponent Random Systems, vol. 6 of Adv. Probab. Related Topics, pages 285–325. Dekker, New York. Translated from the Russian. MR: 599539

Grigorchuk, R.I (1983) On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR, 271(1), 30–33. MR: 85g:20042Google Scholar

Grigor'yan, A.A (1985) The existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds. Mat. Sb. (N.S.), 128(170)(3), 354–363, 446. English translation: Math. USSR-Sb. 56 (1987), no. 2, 349–358. MR: 87d:58140Google Scholar

Grimmett, G. (1999) Percolation, 2nd ed. Springer-Verlag, Berlin. MR: 1707 339

Grimmett, G. (2006) The Random-Cluster Model. Vol. 333 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. MR: 2243761

Grimmett, G. and Kesten, H. (1983) Random electrical networks on complete graphs II: Proofs. Unpublished manuscript, available at http://www.arxiv.org/abs/math.PR/0107068.

Grimmett, G.R, Kesten, H., and Zhang, Y. (1993) Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields, 96(1), 33–44. MR: 94i:60078Google Scholar

Grimmett, G.R and Marstrand, J.M. (1990) The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A, 430(1879), 439–457. MR: 91m:60186Google Scholar

Grimmett, G.R and Newman, C.M. (1990) Percolation in 1 + 1 dimensions. In Grimmett, G.R. and Welsh, D.J.A., editors, Disorder in Physical Systems, pages 167–190. Oxford University Press, New York. A volume in honour of John M. Hammersley on the occasion of his 70th birthday. MR: 92a:60207

Grimmett, G.R and Stacey, A.M. (1998) Critical probabilities for site and bond percolation models. Ann. Probab., 26(4), 1788–1812. MR: 1675079Google Scholar

Grimmett, G.R and Stirzaker, D.R. (2001) Probability and Random Processes, 3rd ed. Oxford University Press, New York. MR: 2059709

Gromov, M. (1981a) Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math., 53, 53–73. MR: 83b:53041Google Scholar

Gromov, M. (1981b) Structures métriques pour les variétés riemanniennes. Vol. 1 of Textes Mathématiques [Mathematical Texts]. CEDIC, Paris. Edited by J., Lafontaine and P., Pansu. MR: 682063

Gromov, M. (1983) Filling Riemannian manifolds. J. Differential Geom., 18(1), 1–147. MR: 697984Google Scholar

Gromov, M. (1999) Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston. Based on the 1981 French original [MR 85e:53051], with appendices by M. Katz, P. Pansu, and S. Semmes, translated from the French by Sean Michael Bates. MR: 2000d:53065

Guillotin-Plantard, N. (2005) Gillis's random walks on graphs. J. Appl. Probab., 42(1), 295–301. MR: 2144913Google Scholar

Guivarc'h, Y. (1980) Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire. In Journées sur les Marches Aléatoires, vol. 74 of Astérisque, pages 47–98, 3. Soc. Math. France, Paris, Paris. Held at Kleebach, March 5–10, 1979. MR: 588157

Guttmann, A.J and Bursill, R.J. (1990) Critical exponent for the loop erased self-avoiding walk by Monte Carlo methods. J. Stat. Phys., 59(1–2), 1–9. http://dx.doi.org/10.1007/BF01015560.Google Scholar

Guttorp, P. (1991) Statistical Inference for Branching Processes. Wiley Series in Probability and Mathematical Statistics. John Wiley, New York. MR: 1254434

Häggström, O. (1994) Aspects of Spatial Random Processes. Ph.D. thesis, Göteborg, Sweden.

Häggström, O. (1995) Random-cluster measures and uniform spanning trees. Stochastic Process. Appl., 59(2), 267–275. MR: 97b:60170Google Scholar

Häggström, O. (1997) Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab., 25(3), 1423–1436. MR: 98f:60207Google Scholar

Häggström, O. (1998) Uniform and minimal essential spanning forests on trees. Random Structures Algorithms, 12(1), 27–50. MR: 99i:05186Google Scholar

Häggström, O. (2013) Two badly behaved percolation processes on a nonunimodular graph. J. Theoret. Probab., 26(4), 1165–1180. MR: 3119989Google Scholar

Häggström, O., Jonasson, J., and Lyons, R. (2002) Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab., 30(1), 443–473. MR: 2003e:60220Google Scholar

Häggström, O. and Peres, Y. (1999) Monotonicity of uniqueness for percolation on Cayley graphs: All infinite clusters are born simultane- ously. Probab. Theory Related Fields, 113(2), 273–285. MR: 1676835Google Scholar

Häggström, O., Peres, Y., and Schonmann, R.H. (1999) Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In Bramson, M. and Durrett, R., editors, Perplexing Problems in Probability, pages 69–90. Birkhäuser, Boston. Festschrift in honor of Harry Kesten. MR: 2000b:60003

Häggström, O., Schonmann, R.H., and Steif, J.E. (2000) The Ising model on diluted graphs and strong amenability. Ann. Probab., 28(3), 1111–1137. MR: 2001i:60169Google Scholar

Hammersley, J.M (1959) Bornes supérieures de la probabilité critique dans un processus de filtration. In Le Calcul des Probabilités et ses Applications. Paris, 15–20 juillet 1958, Colloques Internationaux du Centre National de la Recherche Scientifique, LXXXVII, pages 17–37. Centre National de la Recherche Scientifique, Paris. MR: 0105751

Hammersley, J.M (1961a) Comparison of atom and bond percolation processes. J. Math. Phys., 2, 728–733. MR: 0130722Google Scholar

Hammersley, J.M (1961b) The number of polygons on a lattice. Proc. Cambridge Philos. Soc., 57, 516–523. MR: 23:A814Google Scholar

Han, T.S (1978) Nonnegative entropy measures of multivariate symmetric correlations. Information and Control, 36(2), 133–156. MR: 0464499Google Scholar

Hara, T. and Slade, G. (1990) Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys., 128(2), 333–391. MR: 91a:82037Google Scholar

Hara, T. and Slade, G. (1992) The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys., 4(2), 235–327. MR: 93j:82033Google Scholar

Hara, T. and Slade, G. (1994) Mean-field behaviour and the lace expansion. In Grimmett, G., editor, Probability and Phase Transition, pages 87–122. Kluwer Academic, Dordrecht. Proceedings of the NATO Advanced Study Institute on Probability Theory of Spatial Disorder and Phase Transition held at the University of Cambridge, Cambridge, July 4–16, 1993. MR: 95d:82033

Harmer, G.P and Abbott, D. (1999) Parrondo's paradox. Statist. Sci., 14(2), 206–213. MR: 1722065Google Scholar

de la Harpe, P. and Valette, A. (1989) La propriété de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger). Astérisque, 175, 158. With an appendix by M. Burger. MR: 1023471Google Scholar

Harris, T.E (1952) First passage and recurrence distributions. Trans. Amer. Math. Soc., 73, 471–486. MR: 0052057Google Scholar

Harris, T.E (1960) A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc., 56, 13–20. MR: 22:6023Google Scholar

Hawkes, J. (1970/71) Some dimension theorems for the sample functions of stable processes. Indiana Univ. Math. J., 20, 733–738. MR: 45:1251Google Scholar

Hawkes, J. (1981) Trees generated by a simple branching process. J. London Math. Soc. (2), 24(2), 373–384. MR: 83b:60072Google Scholar

He, Z.X and Schramm, O. (1995) Hyperbolic and parabolic packings. Discrete Comput. Geom., 14(2), 123–149. MR: 1331923Google Scholar

Heathcote, C.R (1966) Corrections and comments on the paper “A branching process allowing immigration”. J. Roy. Statist. Soc. Ser. B, 28, 213–217. MR: 33:1896bGoogle Scholar

Heathcote, C.R, Seneta, E., and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Teor. Verojatnost. i Primenen., 12, 341–346. MR: 36:978Google Scholar

Hebisch, W. and Saloff-Coste, L. (1993) Gaussian estimates for Markov chains and random walks on groups. Ann. Probab., 21(2), 673–709. MR: 1217561Google Scholar

Heyde, C.C (1970) Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Statist., 41, 739–742. MR: 40:8136Google Scholar

Heyde, C.C and Seneta, E. (1977) I. J. Bienaymé. Statistical Theory Anticipated. Vol. 3 of Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York. MR: 57:2855

Higuchi, Y. and Shirai, T. (2003) Isoperimetric constants of regular planar graphs. Interdiscip. Inform. Sci., 9(2), 221–228. MR: 2038013Google Scholar

Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 58, 13–30. MR: 0144363Google Scholar

Hoffman, C., Holroyd, A.E., and Peres, Y. (2006) A stable marriage of Poisson and Lebesgue. Ann. Probab., 34(4), 1241–1272. MR: 2257646Google Scholar

Holopainen, I. and Soardi, P.M. (1997) p-Harmonic functions on graphs and manifolds. Manuscripta Math., 94(1), 95–110. MR: 99c:31017Google Scholar

Holroyd, A.E (2003) Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields, 125(2), 195–224. MR: 1961342Google Scholar

Holroyd, A.E, Levine, L., Mész áros, K., Peres, Y., Propp, J., and Wilson, D.B. (2008) Chip-firing and rotor-routing on directed graphs. In Sidoravicius, V. and Vares, M.E., editors, In and Out of Equilibrium. 2, vol. 60 of Progr. Probab., pages 331–364. Birkhäuser, Basel. Papers from the 10th Brazilian School of Probability (EBP) held in Rio de Janeiro, July 30–August 4, 2006. MR: 2477390

Hoory, S., Linial, N., and Wigderson, A. (2006) Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4), 439–561 (electronic). MR: 2247919Google Scholar

Horn, R.A and Johnson, C.R. (2013) Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge. MR: 2978290

Houdayer, C. (2012) Invariant percolation and measured theory of nonamenable groups [after Gaboriau-Lyons, Ioana, Epstein]. Astérisque, 348, Exp. No. 1039, ix, 339–374. Séminaire Bourbaki: Vol. 2010/2011. Exposés 1027–1042. MR: 3051202Google Scholar

Howard, C.D (2000) Zero-temperature Ising spin dynamics on the homogeneous tree of degree three. J. Appl. Probab., 37(3), 736–747. MR: 1782449Google Scholar

Hutchcroft, T. (2015a) Wired cycle-breaking dynamics for uniform spanning forests. Ann. Probab. To appear, http://www.arxiv.org/abs/1504.03928.

Hutchcroft, T. (2015b) Interlacements and the wired uniform spanning forest. Preprint, http://www.arxiv.org/abs/1512.08509.

Hutchcroft, T. (2016) Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters. C. R. Math. Acad. Sci. Paris, 354(9), 944–947. MR: 3535351Google Scholar

Hutchcroft, T. and Nachmias, A. (2015) Indistinguishability of trees in uniform spanning forests. Probab. Theory Related Fields. To appear.

Hutchcroft, T. and Nachmias, A. (2016) Uniform spanning forests of planar graphs. Preprint, http://www.arxiv.org/abs/1603.07320.

Hutchcroft, T. and Peres, Y. (2015) Boundaries of planar graphs: A unified approach. Preprint, http://www.arxiv.org/abs/1508.03923.

Ichihara, K. (1978) Some global properties of symmetric diffusion processes. Publ. Res. Inst. Math. Sci., 14(2), 441–486. MR: 80d:60099Google Scholar

Imrich, W. (1975) On Whitney's theorem on the unique embeddability of 3-connected planar graphs. In Fiedler, M., editor, Recent Advances in Graph Theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pages 303–306. (Loose errata). Academia, Prague. MR: 52:5462

Indyk, P. and Motwani, R. (1999) Approximate nearest neighbors: Towards removing the curse of dimensionality. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing held in Dallas, TX, May 23–26, 1998, pages 604–613. ACM, New York. MR: 1715608

Jackson, T.S and Read, N. (2010a) Theory of minimum spanning trees. I. Mean-field theory and strongly disordered spin-glass model. Phys. Rev. E, 81(2), 021130. http://dx.doi.org/10.1103/PhysRevE.81.021130.Google Scholar

Jackson, T.S and Read, N. (2010b) Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold. Phys. Rev. E, 81(2), 021131. http://dx.doi.org/10.1103/PhysRevE.81.021131.Google Scholar

Jain, N.C and Taylor, S.J. (1973) Local asymptotic laws for Brownian motion. Ann. Probab., 1, 527–549. MR: 0365732Google Scholar

James, N. and Peres, Y. (1996) Cutpoints and exchangeable events for random walks. Teor. Veroyatnost. i Primenen., 41(4), 854–868. MR: 1687097Google Scholar

Janson, S. (1997) Gaussian Hilbert Spaces. Vol. 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge. MR: 1474726

J árai, A.A. and Redig, F. (2008) Infinite volume limit of the abelian sandpile model in dimensions d3. Probab. Theory Related Fields, 141(1–2), 181–212. MR: 2372969Google Scholar

J árai, A.A. and Werning, N. (2014) Minimal configurations and sandpile measures. J. Theoret. Probab., 27(1), 153–167. MR: 3174221Google Scholar

Jerrum, M. and Sinclair, A. (1989) Approximating the permanent. SIAM J. Comput., 18(6), 1149–1178. MR: 1025467Google Scholar

Joag-Dev, K., Perlman, M.D., and Pitt, L.D. (1983) Association of normal random variables and Slepian's inequality. Ann. Probab., 11(2), 451–455. MR: 690142Google Scholar

Joag-Dev, K. and Proschan, F. (1983) Negative association of random variables, with applications. Ann. Statist., 11(1), 286–295. MR: 85d:62058Google Scholar

Joffe, A. and Waugh, W.A.O'N. (1982) Exact distributions of kin numbers in a Galton-Watson process. J. Appl. Probab., 19(4), 767–775. MR: 84a:60104Google Scholar

Johnson, W.B and Lindenstrauss, J. (1984) Extensions of Lipschitz mappings into a Hilbert space. In Beals, R., Beck, A., Bellow, A., and Hajian, A., editors, Conference in Modern Analysis and Probability, vol. 26 of Contemp. Math., pages 189–206. Amer. Math. Soc., Providence, RI. Held at Yale University, New Haven, Conn., June 8–11, 1982, Held in honor of Professor Shizuo Kakutani. MR: 737400

Jolissaint, P.N and Valette, A. (2014) Lp-distortion and p-spectral gap of finite graphs. Bull. Lond. Math. Soc., 46(2), 329–341. MR: 3194751Google Scholar

Jorgensen, P.ET.and Pearse, E.P.J. (2008) Operator Theory of Electrical Resistance Networks. Universitext. Springer-Verlag. To appear, http://www.arxiv.org/abs/0806.3881.

Joyal, A. (1981) Une théorie combinatoire des séries formelles. Adv. in Math., 42(1), 1–82. MR: 633783Google Scholar

Kahn, J. (2003) Inequality of two critical probabilities for percolation. Electron. Comm. Probab., 8, 184–187 (electronic). MR: 2042 758Google Scholar

Kahn, J., Kim, J.H., Lovász, L., and Vu, V.H. (2000) The cover time, the blanket time, and the Matthews bound. In 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000), pages 467–475. IEEE Comput. Soc. Press, Los Alamitos, CA. MR: 1931843

Kahn, J.D, Linial, N., Nisan, N., and Saks, M.E. (1989) On the cover time of random walks on graphs. J. Theoret. Probab., 2(1), 121–128. MR: 981769Google Scholar

Kaimanovich, V.A (1985) An entropy criterion of maximality for the boundary of random walks on discrete groups. Dokl. Akad. Nauk SSSR, 280(5), 1051–1054. MR: 780288Google Scholar

Kaimanovich, V.A (1990) Boundary and entropy of random walks in random environment. In Grigelionis, B., Sazonov, V.V., and Statulevicius, V., editors, Probability Theory and Mathematical Statistics. Vol. I, pages 573–579. “Mokslas,” Vilnius. Proceedings of the Fifth Conference held in Vilnius, June 25–July 1, 1989. MR: 1153846

Kaimanovich, V.A (1992) Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators. Potential Anal., 1(1), 61–82. MR: 94i:31012Google Scholar

Kaimanovich, V.A (1994) The Poisson boundary of hyperbolic groups. C. R. Acad. Sci. Paris Sér. I Math., 318(1), 59–64. MR: 1260536Google Scholar

Kaimanovich, V.A (1996) Boundaries of invariant Markov operators: The identification problem. In Pollicott, M. and Schmidt, K., editors, Ergodic Theory of Zd Actions, vol. 228 of London Math. Soc. Lecture Note Ser., pages 127–176. Cambridge University Press, Cambridge, Cambridge. Proceedings of the symposium held in Warwick, 1993–1994. MR: 1411218

Kaimanovich, V.A (2000) The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2), 152(3), 659–692. MR: 1815698Google Scholar

Kaimanovich, V.A (2001) Poisson boundary of discrete groups. Preprint, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.6.6675.

Kaimanovich, V.A (2005) “Münchhausen trick” and amenability of self-similar groups. Internat. J. Algebra Comput., 15(5–6), 907–937. MR: 2197814Google Scholar

Kaimanovich, V.A and Masur, H. (1996) The Poisson boundary of the mapping class group. Invent. Math., 125(2), 221–264. MR: 1395719Google Scholar

Kaimanovich, V.A and Masur, H. (1998) The Poisson boundary of Teichmüller space. J. Funct. Anal., 156(2), 301–332. MR: 1636940Google Scholar

Kaimanovich, V.A and Vershik, A.M. (1983) Random walks on discrete groups: Boundary and entropy. Ann. Probab., 11(3), 457–490. MR: 85d:60024Google Scholar

Kaimanovich, V.A and Woess, W. (2002) Boundary and entropy of space homogeneous Markov chains. Ann. Probab., 30(1), 323–363. MR: 1894110Google Scholar

Kakutani, S. (1944) Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo, 20, 706–714. MR: 7,315bGoogle Scholar

Kamae, T. (1982) A simple proof of the ergodic theorem using nonstandard analysis. Israel J. Math., 42(4), 284–290. MR: 682311Google Scholar

Kanai, M. (1985) Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan, 37(3), 391–413. MR: 87d:53082Google Scholar

Kanai, M. (1986) Rough isometries and the parabolicity of Riemannian manifolds. J. Math. Soc. Japan, 38(2), 227–238. MR: 87e:53066Google Scholar

Kargapolov, M.I and Merzljakov, J.I. (1979) Fundamentals of the Theory of Groups. Springer-Verlag, New York. Translated from the second Russian edition by Robert G. Burns. MR: 80k:20002

Karlsson, A. (2003) Boundaries and random walks on finitely generated infinite groups. Ark. Mat., 41(2), 295–306. MR: 2011923Google Scholar

Karlsson, A. and Ledrappier, F. (2007) Linear drift and Poisson boundary for random walks. Pure Appl. Math. Q., 3(4, Special Issue: In honor of Grigory Margulis. Part 1), 1027–1036. MR: 2402595Google Scholar

Karlsson, A. and Woess, W. (2007) The Poisson boundary of lamplighter random walks on trees. Geom. Dedicata, 124, 95–107. MR: 2318539Google Scholar

Kassel, A. and Wilson, D.B. (2016) The looping rate and sandpile density of planar graphs. Amer. Math. Monthly, 123(1), 19–39. MR: 3453533Google Scholar

Kasteleyn, P. (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.Google Scholar

Katznelson, Y. and Weiss, B. (1982) A simple proof of some ergodic theorems. Israel J. Math., 42(4), 291–296. MR: 682312Google Scholar

Kazami, T. and Uchiyama, K. (2008) Random walks on periodic graphs. Trans. Amer. Math. Soc., 360(11), 6065–6087. MR: 2425703Google Scholar

Keane, M. (1995) The essence of the law of large numbers. In Takahashi, Y., editor, Algorithms, Fractals, and Dynamics, pages 125–129. Plenum, New York. Papers from the Hayashibara Forum '92 International Symposium on New Bases for Engineering Science, Algorithms, Dynamics and Fractals held in Okayama, November 23–28, 1992, and the Symposium on Algorithms, Fractals and Dynamics held at Kyoto University, Kyoto, November 30–December 2, 1992. MR: 1402486

Kelly, J.B (1970) Metric inequalities and symmetric differences. In Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), pages 193–212. Academic Press, New York. MR: 0264600

Kemeny, J.G and Snell, J.L. (1960) Finite Markov Chains. The University Series in Undergraduate Mathematics. D. Van, Nostrand, Princeton, NJ. MR: 0115196

Kendall, D.G (1951) Some problems in the theory of queues. J. Roy. Statist. Soc. Ser. B., 13, 151–173; discussion: 173–185. MR: 0047944Google Scholar

Kennelly, A.E (1899) Equivalence of triangles and stars in conducting networks. Electrical World and Engineer, 34, 413–414.Google Scholar

Kenyon, R.W (1997) Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist., 33(5), 591–618. MR: 1473567Google Scholar

Kenyon, R.W (1998) Tilings and discrete Dirichlet problems. Israel J. Math., 105, 61–84. MR: 99m:52026Google Scholar

Kenyon, R.W (2000a) The asymptotic determinant of the discrete Laplacian. Acta Math., 185(2), 239–286. MR: 2002g:82019Google Scholar

Kenyon, R.W (2000b) Long-range properties of spanning trees. J. Math. Phys., 41(3), 1338–1363. MR: 1757 962Google Scholar

Kenyon, R.W (2001) Dominos and the Gaussian free field. Ann. Probab., 29(3), 1128–1137. MR: 1872739Google Scholar

Kenyon, R.W (2008) Height fluctuations in the honeycomb dimer model. Comm. Math. Phys., 281(3), 675–709. MR: 2415464Google Scholar

Kenyon, R.W, Propp, J.G., and Wilson, D.B. (2000) Trees and matchings. Electron. J. Combin., 7(1), Research Paper 25, 34 pp. (electronic). MR: 2001a:05123Google Scholar

Kenyon, R.W and Wilson, D.B. (2015) Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs. J. Amer. Math. Soc., 28(4), 985–1030. MR: 3369907Google Scholar

Kesten, H. (1959a) Full Banach mean values on countable groups. Math. Scand., 7, 146–156. MR: 22:2911Google Scholar

Kesten, H. (1959b) Symmetric random walks on groups. Trans. Amer. Math. Soc., 92, 336–354. MR: 22:253Google Scholar

Kesten, H. (1967) The Martin boundary of recurrent random walks on countable groups. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2, pages 51–74. University of California Press, Berkeley. MR: 0214137

Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals 12. Comm. Math. Phys., 74(1), 41–59. MR: 82c:60179Google Scholar