Project Euclid

References

• [1] 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. 111 505–531.
  Zentralblatt MATH: 0642.60102
  Digital Object Identifier: doi:10.1007/BF01219071
  Project Euclid: euclid.cmp/1104159720
  
• [2] Aldous, D. J. (1990). The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math. 3 450–465.
  Zentralblatt MATH: 0717.05028
  Digital Object Identifier: doi:10.1137/0403039
  
• [3] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.
  Zentralblatt MATH: 1131.60003
  Digital Object Identifier: doi:10.1214/EJP.v12-463
  
• [4] Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87–104.
  Mathematical Reviews (MathSciNet): MR1330762
  Zentralblatt MATH: 0827.60079
  Digital Object Identifier: doi:10.1214/aop/1176988378
  Project Euclid: euclid.aop/1176988378
  
• [5] Babai, L. (1997). The growth rate of vertex-transitive planar graphs. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans, LA, 1997) 564–573. ACM, New York.
  Zentralblatt MATH: 1321.05059
  
• [6] Benjamini, I., Häggström, O. and Schramm, O. (2000). On the effect of adding ϵ-Bernoulli percolation to everywhere percolating subgraphs of ℤ^d. J. Math. Phys. 41 1294–1297.
  Mathematical Reviews (MathSciNet): MR1757959
  Zentralblatt MATH: 0977.82021
  Digital Object Identifier: doi:10.1063/1.533187
  
• [7] Benjamini, I., Jonasson, J., Schramm, O. and Tykesson, J. (2009). Visibility to infinity in the hyperbolic plane, despite obstacles. ALEA Lat. Am. J. Probab. Math. Stat. 6 323–342.
  Zentralblatt MATH: 1276.82012
  
• [8] 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) 160 465–491.
  Zentralblatt MATH: 1071.60006
  Digital Object Identifier: doi:10.4007/annals.2004.160.465
  
• [9] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29–66.
  Zentralblatt MATH: 0924.43002
  Digital Object Identifier: doi:10.1007/s000390050080
  
• [10] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 1347–1356.
  Zentralblatt MATH: 0961.60015
  Digital Object Identifier: doi:10.1214/aop/1022677450
  Project Euclid: euclid.aop/1022677450
  
• [11] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1–65.
  Zentralblatt MATH: 1016.60009
  Digital Object Identifier: doi:10.1214/aop/1008956321
  Project Euclid: euclid.aop/1008956321
  
• [12] Benjamini, I., Lyons, R. and Schramm, O. (1999). Percolation perturbations in potential theory and random walks. In Random Walks and Discrete Potential Theory (Cortona, 1997) 56–84. Cambridge Univ. Press, Cambridge.
  Zentralblatt MATH: 0958.05121
  
• [13] Benjamini, I. and Schramm, O. (1996). Percolation beyond ℤ^d, many questions and a few answers. Electron. Comm. Probab. 1 71–82 (electronic).
  Zentralblatt MATH: 0890.60091
  Digital Object Identifier: doi:10.1214/ECP.v1-978
  
• [14] Benjamini, I. and Schramm, O. (1998). Exceptional planes of percolation. Probab. Theory Related Fields 111 551–564.
  Mathematical Reviews (MathSciNet): MR1641830
  Zentralblatt MATH: 0910.60076
  Digital Object Identifier: doi:10.1007/s004400050177
  
• [15] Benjamini, I. and Schramm, O. (1999). Recent progress on percolation beyond ℤ^d. Microsoft Research. Available at http://research.microsoft.com/en-us/um/people/schramm/pyondrep/.
• [16] Benjamini, I. and Schramm, O. (2001). Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 487–507 (electronic).
  Zentralblatt MATH: 1037.82018
  Digital Object Identifier: doi:10.1090/S0894-0347-00-00362-3
  
• [17] Berger, N. and Biskup, M. (2007). Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 83–120.
  Zentralblatt MATH: 1107.60066
  Digital Object Identifier: doi:10.1007/s00440-006-0498-z
  
• [18] Bollobás, B. and Riordan, O. (2006). The critical probability for random Voronoi percolation in the plane is 1/2. Probab. Theory Related Fields 136 417–468.
• [19] Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge Univ. Press, New York.
  Zentralblatt MATH: 1118.60001
  
• [20] Broder, A. (1989). Genertaing random spanning trees. In 30th Annual Symposium on the Foundations of Computer Science 442–447. IEEE, New York.
• [21] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
  Zentralblatt MATH: 0662.60113
  Digital Object Identifier: doi:10.1007/BF01217735
  Project Euclid: euclid.cmp/1104178143
  
• [22] Burton, R. M. and Keane, M. (1991). Topological and metric properties of infinite clusters in stationary two-dimensional site percolation. Israel J. Math. 76 299–316.
  Zentralblatt MATH: 0757.60102
  Digital Object Identifier: doi:10.1007/BF02773867
  
• [23] De Masi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55 787–855.
  Zentralblatt MATH: 0713.60041
  Digital Object Identifier: doi:10.1007/BF01041608
  
• [24] Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monographs 22. Math. Assoc. America, Washington, DC.
• [25] 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 92 511–527.
  Zentralblatt MATH: 0767.60098
  Digital Object Identifier: doi:10.1007/BF01274266
  
• [26] Grimmett, G. (1999). Percolation, 2nd ed. Springer, Berlin.
  Zentralblatt MATH: 0926.60004
  
• [27] Grimmett, G. (2006). The Random-Cluster Model. Springer, Berlin.
  Zentralblatt MATH: 1122.60087
  
• [28] Grimmett, G. R., Kesten, H. and Zhang, Y. (1993). Random walk on the infinite cluster of the percolation model. Probab. Theory Related Fields 96 33–44.
  Mathematical Reviews (MathSciNet): MR1222363
  Zentralblatt MATH: 0791.60095
  Digital Object Identifier: doi:10.1007/BF01195881
  
• [29] Grimmett, G. R. and Newman, C. M. (1990). Percolation in ∞+1 dimensions. In Disorder in Physical Systems 167–190. Oxford Univ. Press, New York.
  Zentralblatt MATH: 0721.60121
  
• [30] Häggström, O. (1995). Random-cluster measures and uniform spanning trees. Stochastic Process. Appl. 59 267–275.
  Zentralblatt MATH: 0840.60089
  Digital Object Identifier: doi:10.1016/0304-4149(95)00042-6
  
• [31] Häggström, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423–1436.
  Zentralblatt MATH: 0895.60098
  Digital Object Identifier: doi:10.1214/aop/1024404518
  Project Euclid: euclid.aop/1024404518
  
• [32] Häggström, O. (2000). Markov random fields and percolation on general graphs. Adv. in Appl. Probab. 32 39–66.
  Zentralblatt MATH: 0988.60096
  Digital Object Identifier: doi:10.1017/S0001867800009757
  Project Euclid: euclid.aap/1013540021
  
• [33] Häggström, O. (2008). Oded Schramm 1961–2008. Svenska Matematikersamfundets Medlemsutskick 27–29. Available at http://www.math.chalmers.se/~olleh/Oded.pdf.
• [34] Häggström, O. and Jonasson, J. (2006). Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3 289–344 (electronic).
  Zentralblatt MATH: 1189.60175
  Digital Object Identifier: doi:10.1214/154957806000000096
  
• [35] Häggström, O. and Peres, Y. (1999). Monotonicity of uniqueness for percolation on Cayley graphs: All infinite clusters are born simultaneously. Probab. Theory Related Fields 113 273–285.
  Zentralblatt MATH: 0921.60091
  Digital Object Identifier: doi:10.1007/s004400050208
  
• [36] Hara, T. and Slade, G. (1994). Mean-field behaviour and the lace expansion. In Probability and Phase Transition (Cambridge, 1993). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences 420 87–122. Kluwer, Dordrecht.
  Zentralblatt MATH: 0831.60107
  
• [37] Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13–20.
  Zentralblatt MATH: 0122.36403
  Digital Object Identifier: doi:10.1017/S0305004100034241
  
• [38] Harrison, E. R. (1987). Darkness at Night: A Riddle of the Universe. Harvard Univ. Press, Cambridge, MA.
• [39] Kesten, H. (1959). Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 336–354.
  Zentralblatt MATH: 0092.33503
  Digital Object Identifier: doi:10.1090/S0002-9947-1959-0109367-6
  
• [40] Kesten, H. (1959). Full Banach mean values on countable groups. Math. Scand. 7 146–156.
  Mathematical Reviews (MathSciNet): MR112053
  Zentralblatt MATH: 0092.26704
  Digital Object Identifier: doi:10.7146/math.scand.a-10568
  
• [41] Kesten, H. (1980). The critical probability of bond percolation on the square lattice equals ½. Comm. Math. Phys. 74 41–59.
  Mathematical Reviews (MathSciNet): MR575895
  Zentralblatt MATH: 0441.60010
  Digital Object Identifier: doi:10.1007/BF01197577
  Project Euclid: euclid.cmp/1103907931
  
• [42] Kirchhoff, G. (1847). Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72 497–508.
• [43] Lalley, S. P. (1998). Percolation on Fuchsian groups. Ann. Inst. H. Poincaré Probab. Statist. 34 151–177.
  Zentralblatt MATH: 0911.60084
  Digital Object Identifier: doi:10.1016/S0246-0203(98)80022-8
  
• [44] Lalley, S. P. (2001). Percolation clusters in hyperbolic tessellations. Geom. Funct. Anal. 11 971–1030.
  Zentralblatt MATH: 0997.60122
  Digital Object Identifier: doi:10.1007/s00039-001-8223-7
  
• [45] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
• [46] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 237–273.
  Zentralblatt MATH: 1005.60097
  Digital Object Identifier: doi:10.1007/BF02392618
  Project Euclid: euclid.acta/1485891453
  
• [47] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 275–308.
  Mathematical Reviews (MathSciNet): MR1879851
  Zentralblatt MATH: 0993.60083
  Digital Object Identifier: doi:10.1007/BF02392619
  Project Euclid: euclid.acta/1485891454
  
• [48] Lawler, G. F., Schramm, O. and Werner, W. (2002). Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38 109–123.
  Zentralblatt MATH: 1006.60075
  Digital Object Identifier: doi:10.1016/S0246-0203(01)01089-5
  
• [49] Lawler, G. F., Schramm, O. and Werner, W. (2002). Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189 179–201.
  Zentralblatt MATH: 1024.60033
  Digital Object Identifier: doi:10.1007/BF02392842
  
• [50] Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931–958.
  Zentralblatt MATH: 0714.60089
  Digital Object Identifier: doi:10.1214/aop/1176990730
  Project Euclid: euclid.aop/1176990730
  
• [51] Lyons, R. (1992). Random walks, capacity and percolation on trees. Ann. Probab. 20 2043–2088.
  Zentralblatt MATH: 0766.60091
  Digital Object Identifier: doi:10.1214/aop/1176989540
  Project Euclid: euclid.aop/1176989540
  
• [52] Lyons, R. (1998). A bird’s-eye view of uniform spanning trees and forests. In Microsurveys in Discrete Probability (Princeton, NJ, 1997). DIMACS Series in Discrete Mathematics and Theoretical Computer Science 41 135–162. Amer. Math. Soc., Providence, RI.
  Zentralblatt MATH: 0909.60016
  
• [53] Lyons, R. (2000). Phase transitions on nonamenable graphs. J. Math. Phys. 41 1099–1126.
  Zentralblatt MATH: 1034.82014
  Digital Object Identifier: doi:10.1063/1.533179
  
• [54] Lyons, R. (2008). Oded Schramm 1961–2008. IMS Bulletin 37 10–11.
• [55] Lyons, R., Peres, Y. and Schramm, O. (2006). Minimal spanning forests. Ann. Probab. 34 1665–1692.
  Zentralblatt MATH: 1142.60065
  Digital Object Identifier: doi:10.1214/009117906000000269
  Project Euclid: euclid.aop/1163517218
  
• [56] Lyons, R. and Schramm, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809–1836.
  Zentralblatt MATH: 0960.60013
  Digital Object Identifier: doi:10.1214/aop/1022874816
  Project Euclid: euclid.aop/1022874816
  
• [57] Lyons, R. and Schramm, O. (1999). Stationary measures for random walks in a random environment with random scenery. New York J. Math. 5 107–113 (electronic).
  Mathematical Reviews (MathSciNet): MR1703207
  Zentralblatt MATH: 0926.60021
  
• [58] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge Univ. Press, Cambridge.
  Zentralblatt MATH: 0858.60092
  
• [59] Memories of Oded Schramm. Available at http://odedschramm.wordpress.com/.
• [60] Morris, B. (2003). The components of the wired spanning forest are recurrent. Probab. Theory Related Fields 125 259–265.
  Zentralblatt MATH: 1031.60035
  Digital Object Identifier: doi:10.1007/s00440-002-0236-0
  
• [61] Newman, C. M. and Schulman, L. S. (1981). Infinite clusters in percolation models. J. Stat. Phys. 26 613–628.
  Zentralblatt MATH: 0509.60095
  Digital Object Identifier: doi:10.1007/BF01011437
  
• [62] Newman, C. M. and Stein, D. L. (1996). Ground-state structure in a highly disordered spin-glass model. J. Stat. Phys. 82 1113–1132.
  Zentralblatt MATH: 1042.82568
  Digital Object Identifier: doi:10.1007/BF02179805
  
• [63] Pak, I. and Smirnova-Nagnibeda, T. (2000). On non-uniqueness of percolation on nonamenable Cayley graphs. C. R. Acad. Sci. Paris Sér. I Math. 330 495–500.
  Zentralblatt MATH: 0947.43003
  Digital Object Identifier: doi:10.1016/S0764-4442(00)00211-1
  
• [64] Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559–1574.
  Zentralblatt MATH: 0758.60010
  Digital Object Identifier: doi:10.1214/aop/1176990223
  Project Euclid: euclid.aop/1176990223
  
• [65] Peres, Y. (2000). Percolation on nonamenable products at the uniqueness threshold. Ann. Inst. H. Poincaré Probab. Statist. 36 395–406.
  Mathematical Reviews (MathSciNet): MR1770624
  Zentralblatt MATH: 0965.60094
  Digital Object Identifier: doi:10.1016/S0246-0203(00)00130-8
  
• [66] Peres, Y., Pete, G. and Scolnicov, A. (2006). Critical percolation on certain nonunimodular graphs. New York J. Math. 12 1–18 (electronic).
  Zentralblatt MATH: 1103.60079
  
• [67] Russo, L. (1981). On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 229–237.
  Zentralblatt MATH: 0457.60084
  Digital Object Identifier: doi:10.1007/BF00535742
  
• [68] Schonmann, R. H. (1999). Stability of infinite clusters in supercritical percolation. Probab. Theory Related Fields 113 287–300.
  Zentralblatt MATH: 0921.60092
  Digital Object Identifier: doi:10.1007/s004400050209
  
• [69] Schonmann, R. H. (1999). Percolation in ∞+1 dimensions at the uniqueness threshold. In Perplexing Problems in Probability. Progress in Probability 44 53–67. Birkhäuser, Boston, MA.
  Zentralblatt MATH: 0948.60099
  
• [70] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
  Zentralblatt MATH: 0968.60093
  Digital Object Identifier: doi:10.1007/BF02803524
  
• [71] Schramm, O. (2001). A percolation formula. Electron. Comm. Probab. 6 115–120 (electronic).
  Zentralblatt MATH: 1008.60100
  Digital Object Identifier: doi:10.1214/ECP.v6-1041
  
• [72] Schramm, O. (2008). Hyperfinite graph limits. Electron. Res. Announc. Math. Sci. 15 17–23.
  Mathematical Reviews (MathSciNet): MR2372897
  Zentralblatt MATH: 1140.60008
  
• [73] Schramm, O. and Steif, J. E. (2010). Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math. (2) 171 619–672.
  Zentralblatt MATH: 1213.60160
  Digital Object Identifier: doi:10.4007/annals.2010.171.619
  
• [74] Soardi, P. M. (1994). Potential Theory on Infinite Networks. Lecture Notes in Math. 1590. Springer, Berlin.
• [75] Thomassen, C. (1989). Transient random walks, harmonic functions, and electrical currents in infinite electrical networks. Mat-Report 1989-07, Technical Univ. Denmark.
• [76] Timár, Á. (2006). Percolation on nonunimodular transitive graphs. Ann. Probab. 34 2344–2364.
• [77] Trofimov, V. I. (1985). Groups of automorphisms of graphs as topological groups. Mat. Zametki 38 378–385, 476.
• [78] Tykesson, J. (2007). The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space. Electron. J. Probab. 12 1379–1401 (electronic).
  Zentralblatt MATH: 1136.82010
  Digital Object Identifier: doi:10.1214/EJP.v12-460
  
• [79] Werner, W. (2009). Oded Schramm (1961–2008). EMS Newsletter March 9–13.
• [80] Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996) 296–303. ACM, New York.
  Zentralblatt MATH: 0946.60070
  
• [81] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.
  Zentralblatt MATH: 0951.60002