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Percolation and disordered systems

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References

  1. Abraham, D. B., Fontes, L., Newman, C. M., and Piza, M. S. T. (1995). Surface deconstruction and roughening in the multi-ziggurat model of wetting. Physical Review E 52, R1257–R1260.

    Article  Google Scholar 

  2. Abraham, D. B. and Newman, C. M. (1988). Wetting in a three dimensional system: an exact solution. Physical Review Letters 61, 1969–1972.

    Article  MathSciNet  Google Scholar 

  3. Abraham, D. B. and Newman, C. M. (1989). Surfaces and Peierls contours: 3-d wetting and 2-d Ising percolation. Communications in Mathematical Physics 125, 181–200.

    Article  MathSciNet  MATH  Google Scholar 

  4. Abraham, D. B. and Newman, C. M. (1990). Recent exact results on wetting. Wetting Phenomena (J. De Coninck and F. Dunlop, eds.), Lecture Notes in Physics, vol. 354, Springer, Berlin, pp. 13–21.

    Chapter  Google Scholar 

  5. Abraham, D. B. and Newman, C. M. (1991). Remarks on a random surface. Stochastic Orders and Decision under Risk (K. Mosler and M. S. Scarsini, eds.), IMS Lecture Notes, Monograph Series, vol. 19, pp. 1–6.

    Google Scholar 

  6. Abraham, D. B. and Newman, C. M. (1991). The wetting transition in a random surface model. Journal of Statistical Physics 63, 1097–1111.

    Article  MathSciNet  Google Scholar 

  7. Aharony, A. and Stauffer, D. (1991). Introduction to Percolation Theory (Second edition). Taylor and Francis.

    Google Scholar 

  8. Ahlfors, L. V. (1966). Complex Analysis. McGraw-Hill, New York.

    MATH  Google Scholar 

  9. Aizenman, M. (1982). Geometric analysis of φ 4 fields and Ising models. Communications in Mathematical Physics 86, 1–48.

    Article  MathSciNet  MATH  Google Scholar 

  10. Aizenman, M. (1995). The geometry of critical percolation and conformal invariance. Proceedings STATPHYS 1995 (Xianmen) (Hao Bai-lin, ed.), World Scientific.

    Google Scholar 

  11. Aizenman, M. (1996). The critical percolation web: construction and conjectured conformal invariance properties, in preparation.

    Google Scholar 

  12. Aizenman, M. (1997). On the number of incipient spanning clusters. Nuclear Physics B 485, 551–582.

    Article  MathSciNet  MATH  Google Scholar 

  13. Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Communications in Mathematical Physics 108, 489–526.

    Article  MathSciNet  MATH  Google Scholar 

  14. Aizenman, M., Barsky, D. J., and Fernández, R. (1987). The phase transition in a general class of Ising-type models is sharp. Journal of Statistical Physics 47, 343–374.

    Article  MathSciNet  Google Scholar 

  15. Aizenman, M., Chayes, J. T., Chayes, L., Fröhlich, J., and Russo, L. (1983). On a sharp transition from area law to perimeter law in a system of random surfaces. Communications in Mathematical Physics 92, 19–69.

    Article  MathSciNet  MATH  Google Scholar 

  16. Aizenman, M., Chayes, J. T., Chayes, L., and Newman, C. M. (1987). The phase boundary in dilute and random Ising and Potts ferromagnets. Journal of Physics A: Mathematical and General 20, L313.

    Article  MathSciNet  Google Scholar 

  17. Aizenman, M., Chayes, J. T., Chayes, L., and Newman, C. M. (1988). Discontinuity of the magnetization in one-dimensional 1/|x−y|2 Ising and Potts models. Journal of Statistical Physics 50, 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  18. Aizenman, M., and Fernández, R. (1986). On the critical behavior of the magnetization in high-dimensional Ising models. Journal of Statistical Physics 44, 393–454.

    Article  MathSciNet  MATH  Google Scholar 

  19. Aizenman, M. and Fernández, R. (1988). Critical exponents for long-range interactions. Letters in Mathematical Physics 16, 39.

    Article  MathSciNet  MATH  Google Scholar 

  20. Aizenman, M. and Grimmett, G. R. (1991). Strict monotonicity for critical points in percolation and ferromagnetic models. Journal of Statistical Physics 63, 817–835.

    Article  MathSciNet  Google Scholar 

  21. 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. Communications in Mathematical Physics 111, 505–532.

    Article  MathSciNet  MATH  Google Scholar 

  22. Aizenman, M., Kesten, H. and Newman, C. M. (1987). Uniqueness of the infinite cluster and related results in percolation. Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten ed.), IMA Volumes in Mathematics and its Applications, vol. 8, Springer-Verlag, Berlin-Heidelberg-New York, pp. 13–20.

    Chapter  Google Scholar 

  23. Aizenman, M., Klein, A. and Newman, C. M. (1993). Percolation methods for disordered quantum Ising models. Phase Transitions: Mathematics, Physics, Biology… (R. Kotecky, ed.), World Scientific, Singapore, pp. 1–26.

    Google Scholar 

  24. Aizenman, M. and Lebowitz, J. (1988). Metastability in bootstrap percolation. Journal of Physics A: Mathematical and General 21, 3801–3813.

    Article  MathSciNet  MATH  Google Scholar 

  25. Aizenman, M., Lebowitz, J., and Bricmont, J. (1987). Percolation of the minority spins in high-dimensional Ising models. Journal of Statistical Physics 49, 859–865.

    Article  MATH  Google Scholar 

  26. Aizenman, M. and Nachtergaele, B. (1994). Geometric aspects of quantum spin states. Communications in Mathematical Physics 164, 17–63.

    Article  MathSciNet  MATH  Google Scholar 

  27. Aizenman, M. and Newman, C. M. (1984). Tree graph inequalities and critical behavior in percolation models. Journal of Statistical Physics 36, 107–143.

    Article  MathSciNet  MATH  Google Scholar 

  28. Aldous, D. and Steele, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probability Theory and Related Fields 92, 247–258.

    Article  MathSciNet  MATH  Google Scholar 

  29. Alexander, K. S. (1990). Lower bounds on the connectivity function in all directions for Bernoulli percolation in two and three dimensions. Annals of Probability 18, 1547–1562.

    Article  MathSciNet  MATH  Google Scholar 

  30. Alexander, K. S. (1992). Stability of the Wulff construction and fluctuations in shape for large finite clusters in two-dimensional percolation. Probability Theory and Related Fields 91, 507–532.

    Article  MathSciNet  MATH  Google Scholar 

  31. Alexander, K. S. (1993). Finite clusters in high-density continuous percolation: compression and sphericality. Probability Theory and Related Fields 97, 35–63.

    Article  MathSciNet  MATH  Google Scholar 

  32. Alexander, K. S. (1993). A note on some rates of convergence in first-passage percolation. Annals of Applied Probability 3, 81–90.

    Article  MathSciNet  MATH  Google Scholar 

  33. Alexander, K. S. (1995). Simultaneous uniqueness of infinite graphs in stationary random labeled graphs. Communications in Mathematical Physics 168, 39–55.

    Article  MathSciNet  MATH  Google Scholar 

  34. Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Annals of Probability 23, 87–104.

    Article  MathSciNet  MATH  Google Scholar 

  35. Alexander, K. S. (1996). Approximation of subadditive functions and convergence rates in limiting-shape results. Annals of Probability (to appear).

    Google Scholar 

  36. Alexander, K. S. (1996). The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees. Annals of Applied Probability 6, 466–494.

    Article  MathSciNet  MATH  Google Scholar 

  37. Alexander, K. S. (1996). On weak mixing in lattice models (to appear).

    Google Scholar 

  38. Alexander, K. S., Chayes, J. T., and Chayes, L. (1990). The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Communications in Mathematical Physics 131, 1–50.

    Article  MathSciNet  MATH  Google Scholar 

  39. Alexander, K. S. and Molchanov, S. A. (1994). Percolation of level sets for two-dimensional random fields with lattice symmetry. Journal of Statistical Physics 77, 627–643.

    Article  MathSciNet  MATH  Google Scholar 

  40. Andjel, E. (1993). Characteristic exponents for two-dimensional bootstrap percolation. Annals of Probability 21, 926–935.

    Article  MathSciNet  MATH  Google Scholar 

  41. Andjel, E. and Schinazi, R. (1996). A complete convergence theorem for an epidemic model. Journal of Applied Probability 33, 741–748.

    Article  MathSciNet  MATH  Google Scholar 

  42. Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Annals of Probability 24, 1036–1048.

    Article  MathSciNet  MATH  Google Scholar 

  43. Appel, M. J. and Wierman, J. C. (1987). On the absence of AB percolation in bipartite graphs. Journal of Physics A: Mathematical and General 20, 2527–2531.

    Article  MathSciNet  Google Scholar 

  44. Appel, M. J. and Wierman, J. C. (1987). Infinite AB percolation clusters exist on the triangular lattice. Journal of Physics A: Mathematical and General 20, 2533–2537.

    Article  MathSciNet  Google Scholar 

  45. Appel, M. J. and Wierman, J. C. (1993). AB percolation on bond decorated graphs. Journal of Applied Probability 30, 153–166.

    Article  MathSciNet  MATH  Google Scholar 

  46. Ashkin, J. and Teller, E. (1943). Statistics of two-dimensional lattices with four components. The Physical Review 64, 178–184.

    Article  Google Scholar 

  47. Barlow, M. T., Pemantle, R. and Perkins, E. (1997). Diffusion-limited aggregation on a homogenous tree. Probability Theory and Related Fields 107, 1–60.

    Article  MathSciNet  MATH  Google Scholar 

  48. Barsky, D. J. and Aizenman, M. (1991). Percolation critical exponents under the triangle condition. Annals of Probability 19, 1520–1536.

    Article  MathSciNet  MATH  Google Scholar 

  49. Barsky, D. J., Grimmett, G. R. and Newman, C. M. (1991). Dynamic renormalization and continuity of the percolation transition in orthants. Spatial Stochastic Processes (K. S. Alexander and J. C. Watkins, eds.), Birkhäuser, Boston, pp. 37–55.

    Google Scholar 

  50. Barsky, D. J., Grimmett, G. R. and Newman, C. M. (1991). Percolation in half spaces: equality of critical probabilities and continuity of the percolation probability. Probability Theory and Related Fields 90, 111–148.

    Article  MathSciNet  MATH  Google Scholar 

  51. Benjamini, I. (1996). Percolation on groups (to appear).

    Google Scholar 

  52. Benjamini, I. and Kesten, H. (1995). Percolation of arbitrary words in {0, 1}N. Annals of Probability 23, 1024–1060.

    Article  MathSciNet  MATH  Google Scholar 

  53. Benjamini, I., Pemantle, R., and Peres, Y. (1995). Martin capacity for Markov chains. Annals of Probability 23, 1332–1346.

    Article  MathSciNet  MATH  Google Scholar 

  54. Benjamini, I., Pemantle, R., and Peres, Y. (1997). Unpredictable paths and percolation (to appear).

    Google Scholar 

  55. Benjamini, I. and Peres, Y. (1992). Random walks on a tree and capacity in the interval. Annales de l’Institut Henri Poincaré 28, 557–592.

    MathSciNet  MATH  Google Scholar 

  56. Benjamini, I. and Peres, Y. (1994). Tree-indexed random walks on groups and first passage percolation. Probability Theory and Related Fields 98, 91–112.

    Article  MathSciNet  MATH  Google Scholar 

  57. Benjamini, I. and Schramm, O. (1996). Conformal invariance of Voronoi percolation (to appear).

    Google Scholar 

  58. Berg, J. van den (1985). Disjoint occurrences of events: results and conjectures. Particle Systems, Random Media and Large Deviations (R. T. Durrett, ed.), Contemporary Mathematics no. 41, American Mathematical Society, Providence, R. I., pp. 357–361.

    Chapter  Google Scholar 

  59. Berg, J. van den (1993). A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising anti-ferromagnet. Communications in Mathematical Physics 152, 161–166.

    Article  MathSciNet  MATH  Google Scholar 

  60. Berg, J. van den (1996). A constructive mixing condition for 2-D Gibbs measures with random interactions (to appear).

    Google Scholar 

  61. Berg, J. van den (1996). A note on disjoint-occurrence inequalities for marked Poisson point processes. Journal of Applied Probability 33, 420–426.

    Article  MathSciNet  MATH  Google Scholar 

  62. Berg, J. van den (1997). Some reflections on disjoint occurrences of events (to appear).

    Google Scholar 

  63. Berg, J. van den and Ermakov, A. (1996). A new lower bound for the critical probability of site percolation on the square lattice. Random Structures and Algorithms, 199–212.

    Google Scholar 

  64. Berg, J. van den and Fiebig, U. (1987). On a combinatorial conjecture concerning disjoint occurrences of events. Annals of Probability 15, 354–374.

    Article  MathSciNet  MATH  Google Scholar 

  65. Berg, J. van den, Grimmett, G. R., and Schinazi, R. B. (1997). Dependent random graphs and spatial epidemics (to appear).

    Google Scholar 

  66. Berg, J. van den and Keane, M. (1985). On the continuity of the percolation probability function. Particle Systems, Random Media and Large Deviations (R. T. Durrett, ed.), Contemporary Mathematics Series, vol. 26, AMS, Providence, R. I., pp. 61–65.

    Google Scholar 

  67. Berg, J. van den and Kesten, H. (1985). Inequalities with applications to percolation and reliability. Journal of Applied Probability 22, 556–569.

    Article  MathSciNet  MATH  Google Scholar 

  68. Berg, J. van den and Kesten, H. (1993). Inequalities for the time constant in first-passage percolation. Advances in Applied Probability 3, 56–80.

    Article  MathSciNet  MATH  Google Scholar 

  69. Berg, J. van den and Maes, C. (1994). Disagreement percolation in the study of Markov fields. Annals of Probability 22, 749–763.

    Article  MathSciNet  MATH  Google Scholar 

  70. Berg, J. van den and Meester, R. (1991). Stability properties of a flow process in graphs. Random Structures and Algorithms 2, 335–341.

    Article  MathSciNet  MATH  Google Scholar 

  71. Berg, J. van den and Steif, J. (1994). Percolation and the hard-core lattice gas model. Stochastic Processes and their Applications 49, 179–197.

    Article  MathSciNet  MATH  Google Scholar 

  72. Bezuidenhout, C. E. and Grimmett, G. R. (1990). The critical contact process dies out. Annals of Probability 18, 1462–1482.

    Article  MathSciNet  MATH  Google Scholar 

  73. Bezuidenhout, C. E. and Grimmett, G. R. (1991). Exponential decay for subcritical contact and percolation processes. Annals of Probability 19, 984–1009.

    Article  MathSciNet  MATH  Google Scholar 

  74. Bezuidenhout, C. E. and Grimmett, G. R. (1996). A central limit theorem for random walks in random labyrinths (to appear).

    Google Scholar 

  75. Bezuidenhout, C. E., Grimmett, G. R. and Kesten, H. (1993). Strict inequality for critical values of Potts models and random-cluster processes. Communications in Mathematical Physics 158, 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  76. Bezuidenhout, C. E., Grimmett, G. R., and Löffler, A. (1996). Minimal spanning tree analysis of site percolation clusters (to appear).

    Google Scholar 

  77. Bollobás, B., Grimmett, G. R. and Janson, S. (1996). The random-cluster process on the complete graph. Probability Theory and Related Fields 104, 283–317.

    Article  MathSciNet  MATH  Google Scholar 

  78. Borgs, C. and Chayes, J. T. (1996). The covariance matrix of the Potts model: a random cluster analysis. Journal of Statistical Physics 82, 1235–1297.

    Article  MathSciNet  MATH  Google Scholar 

  79. Borgs, C., Chayes, J. T., Kesten, H., and Spencer, J. (1996). The birth of the infinite cluster (to appear).

    Google Scholar 

  80. Bramson, M., Durrett, R. T. and Schonmann, R. H. (1991). The contact process in a random environment. Annals of Probability 19, 960–983.

    Article  MathSciNet  MATH  Google Scholar 

  81. Broadbent, S. R. and Hammersley, J. M. (1957). Percolation processes I. Crystals and mazes. Proceedings of the Cambridge Philosophical Society 53, 629–641.

    Article  MathSciNet  MATH  Google Scholar 

  82. Bunimovitch, L. A. and Troubetzkoy, S. E. (1992). Recurrence properties of Lorentz lattice gas cellular automata. Journal of Statistical Physics 67, 289–302.

    Article  MathSciNet  MATH  Google Scholar 

  83. Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Communications in Mathematical Physics 121, 501–505.

    Article  MathSciNet  MATH  Google Scholar 

  84. Burton, R. M. and Meester, R. (1993). Long range percolation in stationary point processes. Random Structures and Algorithms 4, 177–190.

    Article  MathSciNet  MATH  Google Scholar 

  85. Burton, R. M. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for uniform spanning trees and domino tilings via transfer-impedances. Annals of Probability 21, 1329–1371.

    Article  MathSciNet  MATH  Google Scholar 

  86. Campanino, M., Chayes, J. T. and Chayes, L. (1991). Gaussian fluctuations of connectivities in the subcritical regime of percolation. Probability Theory and Related Fields 88, 269–341.

    Article  MathSciNet  MATH  Google Scholar 

  87. Cardy, J. (1992). Critical percolation in finite geometries. Journal of Physics A: Mathematical and General 25, L201.

    Article  MathSciNet  MATH  Google Scholar 

  88. Chayes, J. T. and Chayes, L. (1987). The mean field bound for the order parameter of Bernoulli percolation, Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten ed.), Springer-Verlag, New York, pp. 49–71.

    Chapter  Google Scholar 

  89. Chayes, J. T. and Chayes, L. (1987). On the upper critical dimension of Bernoulli percolation,. Communications in Mathematical Physics 113, 27–48.

    Article  MathSciNet  MATH  Google Scholar 

  90. Chayes, J. T. and Chayes, L. (1989). The large-N limit of the threshold values in Mandelbrot’s fractal percolation process. Journal of Physics A: Mathematical and General 22, L501–L506.

    Article  MathSciNet  Google Scholar 

  91. Chayes, J. T., Chayes, L. and Durrett, R. T. (1986). Critical behavior of the two-dimensional first passage time. Journal of Statistical Physics 45, 933–948.

    Article  MathSciNet  MATH  Google Scholar 

  92. Chayes, J. T., Chayes, L. and Durrett, R. T. (1988). Connectivity properties of Mandelbrot’s percolation process. Probability Theory and Related Fields 77, 307–324.

    Article  MathSciNet  MATH  Google Scholar 

  93. Chayes, J. T., Chayes, L., Fisher, D. S. and Spencer, T. (1989). Correlation length bounds for disordered Ising ferromagnets. Communications in Mathematical Physics 120, 501–523.

    Article  MathSciNet  MATH  Google Scholar 

  94. Chayes, J. T., Chayes, L., Grannan, E. and Swindle, G. (1991). Phase transitions in Mandelbrot’s percolation process in 3 dimensions. Probability Theory and Related Fields 90, 291–300.

    Article  MathSciNet  MATH  Google Scholar 

  95. Chayes, J. T., Chayes, L., Grimmett, G. R., Kesten, H. and Schonmann, R. H. (1989). The correlation length for the high density phase of Bernoulli percolation. Annals of Probability 17, 1277–1302.

    Article  MathSciNet  MATH  Google Scholar 

  96. Chayes, J. T., Chayes, L. and Kotecky, R. (1995). The analysis of the Widom-Rowlinson model by stochastic geometric methods. Communications in Mathematical Physics 172, 551–569.

    Article  MathSciNet  MATH  Google Scholar 

  97. Chayes, J. T., Chayes, L. and Newman, C. M. (1987). Bernoulli percolation above threshold: an invasion percolation analysis. Annals of Probability 15, 1272–1287.

    Article  MathSciNet  MATH  Google Scholar 

  98. Chayes, J. T., Chayes, L. and Schonmann, R. H. (1987). Exponential decay of connectivities in the two dimensional Ising model. Journal of Statistical Physics 49, 433–445.

    Article  MathSciNet  MATH  Google Scholar 

  99. Chayes, L. (1991). On the critical behavior of the first passage time in d=3. Helvetica Physica Acta 64, 1055–1069.

    MathSciNet  Google Scholar 

  100. Chayes, L. (1993). The density of Peierls contours in d=2 and the height of the wedding cake. Journal of Physics A: Mathematical and General 26, L481–L488.

    Article  MathSciNet  MATH  Google Scholar 

  101. Chayes, L. (1995). On the absence of directed fractal percolation. Journal of Physics A: Mathematical and General 28, L295–L301.

    Article  MathSciNet  MATH  Google Scholar 

  102. Chayes, L. (1995). Aspects of the fractal percolation process. Fractal Geometry and Stochastics (C. Bandt, S. Graf and M. Zähle, eds.), Birkhäuser, Boston, pp. 113–143.

    Chapter  Google Scholar 

  103. Chayes, L. (1996). On the length of the shortest crossing in the super-critical phase of Mandelbrot’s percolation process. Stochastic Processes and their Applications 61, 25–43.

    Article  MathSciNet  MATH  Google Scholar 

  104. Chayes, L. (1996). Percolation and ferromagnetism on Z 2: the q-state Potts cases (to appear).

    Google Scholar 

  105. Chayes, L., Kotecky, R. and Shlosman, S. B. (1995). Aggregation and intermediate phases in dilute spin-systems. Communications in Mathematical Physics 171, 203–232.

    Article  MathSciNet  MATH  Google Scholar 

  106. Chayes, L. and Winfield, C. (1993). The density of interfaces: a new first passage problem. Journal of Applied Probability 30, 851–862.

    Article  MathSciNet  MATH  Google Scholar 

  107. Chow, Y. S. and Teicher, H. (1978). Probability Theory. Springer, Berlin.

    Book  MATH  Google Scholar 

  108. Clifford, P. (1990). Markov random fields in statistics. Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh, eds.), Oxford University Press, Oxford, pp. 19–32.

    Google Scholar 

  109. Cohen, E. G. D. (1991). New types of diffusions in lattice gas cellular automata. Microscopic Simulations of Complex Hydrodynamical Phenomena (M. Mareschal and B. L. Holian eds.), Plenum Press, New York, pp. 137–152.

    Google Scholar 

  110. Cohen, E. G. D. and Wang, F. (1995). New results for diffusion in Lorentz lattice gas cellular automata. Journal of Statistical Physics 81, 445–466.

    Article  MATH  Google Scholar 

  111. Cohen, E. G. D. and Wang, F. (1995). Novel phenomena in Lorentz lattice gases. Physica A 219, 56–87.

    Article  Google Scholar 

  112. Cox, J. T., Gandolfi, A., Griffin, P. and Kesten, H. (1993). Greedy lattice animals I: Upper bounds. Advances in Applied Probability 3, 1151–1169.

    Article  MathSciNet  MATH  Google Scholar 

  113. Dekking, F. M. and Grimmett, G. R. (1988). Superbranching processes and projections of random Cantor sets. Probability Theory and Related Fields 78, 335–355.

    Article  MathSciNet  MATH  Google Scholar 

  114. Dekking, F. M. and Meester, R. W. J. (1990). On the structure of Mandelbrot’s percolation process and other random Cantor sets. Journal of Statistical Physics 58, 1109–1126.

    Article  MathSciNet  MATH  Google Scholar 

  115. DeMasi, A., Ferrari, P. A., Goldstein, S. and Wick, W. D. (1985). Invariance principle for reversible Markov processes with application to diffusion in the percolation regime. Particle Systems, Random Media and Large Deviations (R. T. Durrett ed.), Contemporary Mathematics, no. 41, American Mathematical Society, Providence, R. I., pp. 71–85.

    Chapter  Google Scholar 

  116. DeMasi, 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. Journal of Statistical Physics 55, 787–855.

    Article  MathSciNet  MATH  Google Scholar 

  117. Deuschel, J.-D. and Pisztora, A. (1996). Surface order large deviations for high-density percolation. Probability Theory and Related Fields 104, 467–482.

    Article  MathSciNet  MATH  Google Scholar 

  118. Doyle, P. G. and Snell, E. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monograph, no. 22. AMA, Washington, D. C..

    MATH  Google Scholar 

  119. Durrett, R. T. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks/Cole, Pacific Grove, California.

    MATH  Google Scholar 

  120. Durrett, R. T. (1991). The contact process, 1974–1989. Mathematics of Random Media Blackburg, VA, Lectures in Applied Mathematics, vol. 27, AMS, pp. 1–18.

    MathSciNet  MATH  Google Scholar 

  121. Durrett, R. T. (1995). Ten lectures on particle systems. Ecole d’Eté de Probabilités de Saint Flour XXIII-1993 (P. Bernard, ed.), Lecture Notes in Mathematics no. 1608, Springer, Berlin, pp. 97–201.

    Google Scholar 

  122. Durrett, R. T. and Kesten, H. (1990). The critical parameter for connectedness of some random graphs. A Tribute to Paul Erdős (A. Baker, B. Bollobás, and A. Hajnal eds.), Cambridge University Press, Cambridge pp. 161–176.

    Chapter  Google Scholar 

  123. Durrett, R. T. and Neuhauser, C. (1991). Epidemics with recovery. Advances in Applied Probability 1, 189–206.

    Article  MathSciNet  MATH  Google Scholar 

  124. Durrett, R. T. and Schonmann, R. H. (1988). The contact process on a finite set II. Annals of Probability 16, 1570–1583.

    Article  MathSciNet  MATH  Google Scholar 

  125. Durrett, R. T. and Schonmann, R. H. (1988). Large deviations for the contact process and two dimensional percolation. Probability Theory and Related Fields 77, 583–603.

    Article  MathSciNet  MATH  Google Scholar 

  126. Durrett, R. T., Schonmann, R. H. and Tanaka, N. I. (1989). The contact process on a finite set III. The critical case. Annals of Probability 17, 1303–1321.

    Article  MathSciNet  MATH  Google Scholar 

  127. Durrett, R. T., Schonmann, R. H. and Tanaka, N. I. (1989). Correlation lengths for oriented percolation. Journal of Statistical Physics 55, 965–979.

    Article  MathSciNet  MATH  Google Scholar 

  128. Edwards, R. G. and Sokal, A. D. (1988). Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. The Physical Review D 38, 2009–2012.

    Article  MathSciNet  Google Scholar 

  129. Ehrenfest, P. (1959). Collected Scientific Papers (M. J. Klein, ed.). North-Holland, Amsterdam.

    Google Scholar 

  130. Enter, A. C. D. van and Hollander, F. den (1993). Interacting particle systems and Gibbs measure, Lecture notes for Master Class 1992–1993, Mathematical Research Institute, The Netherlands.

    Google Scholar 

  131. Falconer, K. J. (1986). Random fractals. Mathematical Proceedings of the Cambridge Philosophical Society 100, 559–582.

    Article  MathSciNet  MATH  Google Scholar 

  132. Falconer, K. J. (1986). Sets with prescribed projections and Nikodym sets. Proceedings of the London Mathematical Society 53, 48–64.

    Article  MathSciNet  MATH  Google Scholar 

  133. Falconer, K. J. (1989). Projections of random Cantor sets. Journal of Theoretical Probability 2, 65–70.

    Article  MathSciNet  MATH  Google Scholar 

  134. Falconer, K. J. and Grimmett, G. R. (1992). On the geometry of random Cantor sets and fractal percolation. Journal of Theoretical Probability 5, 465–485; Erratum (1994) 7, 209–210.

    Article  MathSciNet  MATH  Google Scholar 

  135. Fernández, R., Fröhlich, J. and Sokal, A. D. (1992). Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer-Verlag, Berlin.

    Book  MATH  Google Scholar 

  136. Fill, J. and Pemantle, R. (1992). Oriented percolation, first-passage percolation and covering times for Richardson’s model on the n-cube. Annals of Applied Probability 3, 593–629.

    Article  MathSciNet  MATH  Google Scholar 

  137. Fiocco, M. and Zwet, W. R. van (1996). Consistent estimation for the supercritical contact process (to appear).

    Google Scholar 

  138. Fisher, M. E. (1961). Critical probabilities for cluster size and percolation problems. Journal of Mathematical Physics 2, 620–627.

    Article  MathSciNet  MATH  Google Scholar 

  139. Fontes, L. and Newman, C. M. (1993). First passage percolation for random colorings of Z d. Annals of Applied Probability 3, 746–762; Erratum 4, 254.

    Article  MathSciNet  MATH  Google Scholar 

  140. Fortuin, C. M. (1971). On the random-cluster model. Doctoral thesis, University of Leiden.

    Google Scholar 

  141. Fortuin, C. M. (1972). On the random cluster model. II. The percolation model. Physica 58, 393–418.

    Article  MathSciNet  Google Scholar 

  142. Fortuin, C. M. (1972). On the random cluster model. III. The simple random-cluster process. Physica 59, 545–570.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  144. Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Communications in Mathematical Physics 22, 89–103.

    Article  MathSciNet  MATH  Google Scholar 

  145. Friedgut, E. and Kalai, G. (1996). Every monotone graph property has a sharp threshold. Proceedings of the American Mathematical Society 124, 2993–3002.

    Article  MathSciNet  MATH  Google Scholar 

  146. Gandolfi, A., Grimmett, G. R. and Russo, L. (1988). On the uniqueness of the infinite open cluster in the percolation model. Communications in Mathematical Physics 114, 549–552.

    Article  MathSciNet  MATH  Google Scholar 

  147. Gandolfi, A., Keane, M. and Newman, C. M. (1992). Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probability Theory and Related Fields 92, 511–527.

    Article  MathSciNet  MATH  Google Scholar 

  148. Gandolfi, A. and Kesten, H. (1994). Greedy lattice animals II: Linear growth. Advances in Applied Probability 4, 76–107.

    Article  MathSciNet  MATH  Google Scholar 

  149. Gandolfi, A., Newman, C. M. and Stein, D. L. (1993). Exotic states in long range spin glasses. Communications in Mathematical Physics 157, 371–387.

    Article  MathSciNet  MATH  Google Scholar 

  150. Georgii, H.-O. (1988). Gibbs Measures and Phase transitions. Walter de Gruyter, Berlin.

    Book  MATH  Google Scholar 

  151. Gordon, D. M. (1991). Percolation in high dimensions. Bulletin of the London Mathematical Society 44, 373–384.

    Article  MathSciNet  MATH  Google Scholar 

  152. Graf, S. (1987). Statistically self-similar fractals. Probability Theory and Related Fields 74, 357–392.

    Article  MathSciNet  MATH  Google Scholar 

  153. Graf, S., Mauldin, R. D. and Williams, S. C. (1987). Exact Hausdorff dimension in random recursive constructions. Memoris of the American Mathematical Society 381.

    Google Scholar 

  154. Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Mathematics, vol. 724, Springer, Berlin.

    MATH  Google Scholar 

  155. Grimmett, G. R. (1978). The rank functions of large random lattices. Journal of London Mathematical Society 18, 567–575.

    Article  MathSciNet  MATH  Google Scholar 

  156. Grimmett, G. R. (1989). Percolation. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  157. Grimmett, G. R. (1993). Differential inequalities for Potts and random-cluster processes. Cellular Automata and Cooperative Systems (N. Boccara et al., eds.), Kluwer, Dordrecht, pp. 227–236.

    Chapter  Google Scholar 

  158. Grimmett, G. R. (1994). Potts models and random-cluster processes with many-body interactions. Journal of Statistical Physics 75, 67–121.

    Article  MathSciNet  MATH  Google Scholar 

  159. Grimmett, G. R. (1994). The random-cluster model. Probability, Statistics and Optimisation (F. P. Kelly, ed.), John Wiley & Sons, Chichester, pp. 49–63.

    Google Scholar 

  160. Grimmett, G. R. (1994). Percolative problems. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 69–86.

    Chapter  Google Scholar 

  161. Grimmett, G. R. (1994). Probability and Phase Transition. editor. Kluwer, Dordrecht.

    Book  MATH  Google Scholar 

  162. Grimmett, G. R. (1995). Comparison and disjoint-occurrence inequalities for random-cluster models. Journal of Statistical Physics 78, 1311–1324.

    Article  MathSciNet  MATH  Google Scholar 

  163. Grimmett, G. R. (1995). The stochastic random-cluster process and the uniqueness of random-cluster measures. Annals of Probability 23, 1461–1510.

    Article  MathSciNet  MATH  Google Scholar 

  164. Grimmett, G. R., Kesten, H. and Zhang, Y. (1993). Random walk on the infinite cluster of the percolation model. Probability Theory and Related Fields 96, 33–44.

    Article  MathSciNet  MATH  Google Scholar 

  165. Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proceedings of the Royal Society (London), Series A 430, 439–457.

    Article  MathSciNet  MATH  Google Scholar 

  166. Grimmett, G. R., Menshikov, M. V. and Volkov, S. E. (1996). Random walks in random labyrinths. Markov Processes and Related Fields 2, 69–86.

    MathSciNet  MATH  Google Scholar 

  167. Grimmett, G. R. and Newman, C. M. (1990). Percolation in ∞+1 dimensions. Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh eds.), Clarendon Press, Oxford, pp. 167–190.

    Google Scholar 

  168. Grimmett, G. R. and Piza, M. S. T. (1996). Decay of correlations in subcritical Potts and random-cluster models (to appear).

    Google Scholar 

  169. Grimmett, G. R. and Stacey, A. M. (1996). Inequalities for critical probabilities of site and bond percolation (to appear).

    Google Scholar 

  170. Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes. Oxford University Press, Oxford.

    MATH  Google Scholar 

  171. Häggström, O. (1996). The random-cluster model on a homogeneous tree. Probability Theory and Related Fields 104, 231–253.

    Article  MathSciNet  MATH  Google Scholar 

  172. Häggström, O. and Pemantle, R. (1995). First-passage percolation and a model for competing spatial growth (to appear).

    Google Scholar 

  173. Häggström. O., Peres, Y., and Steif, J. (1995). Dynamical percolation (to appear).

    Google Scholar 

  174. Hammersley, J. M. (1957). Percolation processes. Lower bounds for the critical probability. Annasl of Mathematical Statistics 28, 790–795.

    Article  MathSciNet  MATH  Google Scholar 

  175. Hammersley, J. M. (1959). Bornes supérieures de la probabilité critique dans un processus de filtration. Le Calcul de Probabilité et ses Applications, CNRS, Paris, pp. 17–37.

    MATH  Google Scholar 

  176. Hara, T. and Slade, G. (1989). The mean-field critical behaviour of percolation in high dimensions. Proceedings of the IXth International Congress on Mathematical Physics (B. Simon, A. Truman, I. M. Davies, eds.), Adam Hilger, Bristol, pp. 450–453.

    Google Scholar 

  177. Hara, T. and Slade, G. (1989). The triangle condition for percolation. Bulletin of the American Mathematical Society 21, 269–273.

    Article  MathSciNet  MATH  Google Scholar 

  178. Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Communications in Mathematical Physics 128, 333–391.

    Article  MathSciNet  MATH  Google Scholar 

  179. Hara, T. and Slade, G. (1994). Mean-field behaviour and the lace expansion. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 87–122.

    Chapter  Google Scholar 

  180. Hara, T. and Slade, G. (1995). The self-avoiding walk and percolation critical points in high dimensions. Combinatorics, Probability, Computing 4, 197–215.

    Article  MathSciNet  MATH  Google Scholar 

  181. Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proceedings of the Cambridge Philosophical Society 56, 13–20.

    Article  MathSciNet  MATH  Google Scholar 

  182. Hauge, E. H. and Cohen, E. G. D. (1967). Normal and abnormal effects in Ehrenfest wind-tree model.. Physics Letters 25A, 78–79.

    Article  Google Scholar 

  183. Hawkes, J. (1981). Trees generated by a simple branching process. Journal of the London Mathematical Society 24, 373–384.

    Article  MathSciNet  MATH  Google Scholar 

  184. Higuchi, Y. (1989). A remark on the percolation for the 2D Ising model. Osaka Journal of Mathematics 26, 207–224.

    MathSciNet  MATH  Google Scholar 

  185. Higuchi, Y. (1991). Level set representation for the Gibbs state of the ferromagnetic Ising model. Probability Theory and Related Fields 90, 203–221.

    Article  MathSciNet  MATH  Google Scholar 

  186. Higuchi, Y. (1992). Percolation (in Japanese). Yuseisha/Seiunsha.

    Google Scholar 

  187. Higuchi, Y. (1993). A sharp transition for the two-dimensional Ising percolation. Probability Theory and Related Fields 97, 489–514.

    Article  MathSciNet  MATH  Google Scholar 

  188. Higuchi, Y. (1993). Coexistence of infinite *-clusters II: Ising percolation in two dimensions. Probability Theory and Related Fields 97, 1–34.

    Article  MathSciNet  MATH  Google Scholar 

  189. Higuchi, Y. (1996). Ising model percolation. Sugaku Expositions, vol. 47, pp. 111–124.

    MathSciNet  Google Scholar 

  190. Higuchi, Y. (1996). Mixing condition and percolation for Ising models (to appear).

    Google Scholar 

  191. Hintermann, D., Kunz, H. and Wu, F. Y. (1978). Exact results for the Potts model in two dimensions. Journal of Statistical Physics 19, 623–632.

    Article  MathSciNet  Google Scholar 

  192. Holley, R. (1974). Remarks on the FKG inequalities. Communications in Mathematical Physics 36, 227–231.

    Article  MathSciNet  Google Scholar 

  193. Imbrie, J. and Newman, C. M. (1988). An intermediate phase with slow decay of correlations in one dimensional 1/∣x−y∣2 percolation, Ising and Potts models. Communications in Mathematical Physics 118, 303–336.

    Article  MathSciNet  Google Scholar 

  194. Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik 31, 253–258.

    Article  Google Scholar 

  195. Jacobs, D. J. and Thorpe, M. F. (1995). Generic rigidity percolation: the pebble game. Physical Review Letters 75, 4051–4054.

    Article  Google Scholar 

  196. Jacobs, D. J. and Thorpe, M. F. (1996). Rigidity percolation in two dimensions. The pebble game. Physical Review E 53, 3682–3693.

    Article  Google Scholar 

  197. Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, Chichester.

    MATH  Google Scholar 

  198. Kantor, T. and Hassold, G. N. (1988). Topological entanglements in the percolation problem. Physical Review Letters 60, 1457–1460.

    Article  MathSciNet  Google Scholar 

  199. Kasteleyn, P. W. and Fortuin, C. M. (1969). Phase transitions in lattice systems with random local properties. Journal of the Physical Society of Japan 26, 11–14, Supplement.

    Google Scholar 

  200. Kesten, H. (1980). The critical probability of bond percolation on the square lattice equals 1/2. Communications in Mathematical Physics 74, 41–59.

    Article  MathSciNet  MATH  Google Scholar 

  201. Kesten, H. (1980). On the time constant and path length of first-passage percolation. Advances in Applied Probability 12, 848–863.

    Article  MathSciNet  MATH  Google Scholar 

  202. Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.

    Book  MATH  Google Scholar 

  203. Kesten, H. (1986). Aspects of first-passage percolation. Ecole d’Eté de Probabilités de Saint Flour XIV-1984 (P. L. Hennequin, ed.), Lecture Notes in Mathematics no. 1180, Springer, Berlin, pp. 125–264.

    Chapter  Google Scholar 

  204. Kesten, H. (1987). Scaling relations for 2D-percolation. Communications in Mathematical Physics 109, 109–156.

    Article  MathSciNet  MATH  Google Scholar 

  205. Kesten, H. (1987). A scaling relation at criticality for 2D-percolation. Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten, ed.), Springer, Berlin, pp. 203–212.

    Chapter  Google Scholar 

  206. Kesten, H. (1987). Percolation theory and first-passage percolation. Annals of Probability 15, 1231–1271.

    Article  MathSciNet  MATH  Google Scholar 

  207. Kesten, H. (1988). Recent results in rigorous percolation theory. Astérisque 157–158.

    Google Scholar 

  208. Kesten, H. (1988). Correlation length and critical probabilities of slabs for percolation, preprint.

    Google Scholar 

  209. Kesten, H. (1990). Asymtotics in high dimensions for percolation. Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh eds.), Clarendon Press, Oxford.

    Google Scholar 

  210. Kesten, H. (1991). Asymptotics in high dimensions for the Fortuin-Kasteleyn random cluster mode, Spatial Stochastic Processes (K. S. Alexander and J. C. Watkins eds.), Birkhäuser, Boston, pp. 57–85.

    Chapter  Google Scholar 

  211. Kesten, H. (1992). Connectivity of certain graphs on halfspaces, quarter spaces, …. Probability Theory (L. H. Y. Chen, K. P. Choi, K. Hu, and J.-H. Lou, eds.), Walter de Gruyter, Berlin, pp. 91–104.

    Google Scholar 

  212. Kesten, H. (1992). First-and last-passage percolation. Proceedings of Statistics Conference in Rio de Janeiro, Brasil.

    Google Scholar 

  213. Kesten, H. and Schonmann, R. H. (1990). Behavior in large dimensions of the Potts and Heisenberg models. Reviews in Mathematical Physics 1, 147–182.

    Article  MathSciNet  MATH  Google Scholar 

  214. Kesten, H. and Zhang, Y. (1987). Strict inequalities for some critical exponents in two-dimensional percolation. Journal of Statistical Physics 46, 1031–1055.

    Article  MathSciNet  MATH  Google Scholar 

  215. Kesten, H. and Zhang, Y. (1990). The probability of a large finite cluster in supercritical Bernoulli percolation. Annals of Probability 18, 537–555.

    Article  MathSciNet  MATH  Google Scholar 

  216. Kesten, H. and Zhang, Y. (1993). The tortuosity of occupied crossing of a box in critical percolation. Journal of Statistical Physics 70, 599–611.

    Article  MathSciNet  MATH  Google Scholar 

  217. Kesten, H. and Zhang, Y. (1997). A central limit theorem for ‘critical’ first passage percolation in two dimensions. Probability Theory and Related Fields 107, 137–160.

    Article  MathSciNet  MATH  Google Scholar 

  218. Kihara, T., Midzuno, Y. and Shizume, J. (1954). Statistics of two-dimensional lattices with many components. Journal of the Physical Society of Japan 9, 681–687.

    Article  Google Scholar 

  219. Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion. Communications in Mathematical Physics 104, 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  220. Klein, A. (1994). Multiscale analysis in disordered systems: Percolation and contact process in a random environment. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 139–152.

    Chapter  Google Scholar 

  221. Klein, A. (1994). Extinction of contact and percolation processes in a random environment. Annals of Probability 22, 1227–1251.

    Article  MathSciNet  MATH  Google Scholar 

  222. Kotecký, R. (1994). Geometric representation of lattice models and large volume asymptotics. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 153–176.

    Chapter  Google Scholar 

  223. Kotecký, R. and Shlosman, S. (1982). First order phase transitions in large entropy lattice systems. Communications in Mathematical Physics 83, 493–515.

    Article  MathSciNet  Google Scholar 

  224. Kuulasmaa, K. (1982). The spatial general epidemic and locally dependent random graphs. Journal of Applied Probability 19, 745–758.

    Article  MathSciNet  MATH  Google Scholar 

  225. Laanait, L., Messager, A., Miracle-Sole, S., Ruiz, J. and Shlosman, S. (1991). Interfaces in the Potts model I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation. Communications in Mathematical Physics 140, 81–91.

    Article  MathSciNet  MATH  Google Scholar 

  226. Laanait, L., Messager, A. and Ruiz, J. (1986). Phase coexistence and surface tensions for the Potts model. Communications in Mathematical Physics 105, 527–545.

    Article  MathSciNet  Google Scholar 

  227. Langlands, R. P. (1993). Dualität bei endlichen Modellen der Perkolation. Mathematische Nachrichten 160, 7–58.

    Article  MathSciNet  Google Scholar 

  228. Langlands, R. P. (1996). An essay on the dynamics and statistics of critical field theories. 50th anniversary volume of the Canadian Mathematical Society (to appear).

    Google Scholar 

  229. Langlands, R. P. and Lafortune, M.-A. (1994). Finite models for percolation. Contemporary Mathematics 177, 227–246.

    Article  MathSciNet  MATH  Google Scholar 

  230. Langlands, R., Pichet, C., Pouliot, P. and Saint-Aubin, Y. (1992). On the universality of crossing probabilities in two-dimensional percolation. Journal of Statistical Physics 67, 553–574.

    Article  MathSciNet  MATH  Google Scholar 

  231. Langlands, R., Pouliot, P. and Saint-Aubin, Y. (1994). Conformal invariance in two-dimensional percolation. Bulletin of the American Mathematical Society 30, 1–61.

    Article  MathSciNet  MATH  Google Scholar 

  232. Lebowitz, J. and Martin-Löf, A. (1972). On the uniqueness of the equilibrium state for Ising spin systems. Communications in Mathematical Physics 25, 276–282.

    Article  MathSciNet  Google Scholar 

  233. Licea, C. and Newman, C. M. (1996). Geodesics in two-dimensional first-passage percolation. Annals of Probability 24, 399–410.

    Article  MathSciNet  MATH  Google Scholar 

  234. Licea, C., Newman, C. M., and Piza, M. S. T. (1995). Superdiffusivity in first-passage percolation (to appear).

    Google Scholar 

  235. Lieb, E. H. (1980). A refinement of Simon’s correlation inequality. Communications in Mathematical Physics 77, 127–135.

    Article  MathSciNet  Google Scholar 

  236. Liggett, T. M. (1985). Interacting Particle Systems. Springer-Verlag, Berlin.

    Book  MATH  Google Scholar 

  237. Liggett, T. M. (1991). Spatially inhomogeneous contact processes. Spatial Stochastic Processes (K. S. Alexander and J. C. Watkins, eds.), Birkhäuser, Boston, pp. 105–140.

    Chapter  Google Scholar 

  238. Liggett, T. M. (1991). The periodic threshold contact process. Random Walks, Brownian Motion and Interacting Particle Systems (R. T. Durrett and H. Kesten, eds.), Birkhäuser, Boston, pp. 339–358.

    Chapter  Google Scholar 

  239. Liggett, T. M. (1992). The survival of one-dimensional contact processes in random environments. Annals of Probability 20, 696–723.

    Article  MathSciNet  MATH  Google Scholar 

  240. Ligget, T. M. (1994). Coexistence in threshold voter models. Annals of Probability 22, 764–802.

    Article  MathSciNet  MATH  Google Scholar 

  241. Liggett, T. M. (1995). Improved upper bounds for the contact process critical value. Annals of Probability 23, 697–723.

    Article  MathSciNet  MATH  Google Scholar 

  242. Liggett, T. M. (1995). Survival of discrete time growth models, with applications to oriented percolation. Annals of Applied Probability 5, 613–636.

    Article  MathSciNet  MATH  Google Scholar 

  243. Liggett, T. M. (1996). Multiple transition points for the contact process on the binary tree. Annals of Probability 24, 1675–1710.

    Article  MathSciNet  MATH  Google Scholar 

  244. Liggett, T. M., Schonmann, R. H., and Stacey, A. (1997). Domination by product measures. Annals of Probability 25 (to appear).

    Google Scholar 

  245. Lorentz, H. A. (1905). The motion of electrons in metallic bodies, I, II, and III. Koninklijke Akademie van Wetenschappen te Amsterdam, Section of Sciences 7, 438–453, 585–593, 684–691.

    Google Scholar 

  246. Łuczak, T. and Wierman, J. C. (1989). Critical probability bounds for two-dimensional site percolation models. Journal of Physics A: Mathematical and General 21, 3131–3138.

    Article  MathSciNet  MATH  Google Scholar 

  247. Łuczak, T. and Wierman, J. C. (1989). Counterexamples in AB percolation. Journal of Physics A: Mathematical and General 22, 185–191.

    Article  MathSciNet  MATH  Google Scholar 

  248. Lyons, R. (1990). Random walks and percolation on trees. Annals of Probability 18, 931–958.

    Article  MathSciNet  MATH  Google Scholar 

  249. Lyons, R. (1992). Random walks, capacity and percolation on trees. Annals of Probability 20, 2043–2088.

    Article  MathSciNet  MATH  Google Scholar 

  250. Lyons, R. and Pemantle, R. (1992). Random walk in a random environment and first-passage percolation on trees. Annals of Probability 20, 125–137.

    Article  MathSciNet  MATH  Google Scholar 

  251. Lyons, T. (1983). A simple criterion for transience of a reversible Markov chain. Annals of Probability 11, 393–402.

    Article  MathSciNet  MATH  Google Scholar 

  252. Madras, N., Schinazi, R. B., and Schonmann, R. H. (1994). On the critical behavior of the contact process in deterministic inhomogeneous environment. Annals of Probability 22, 1140–1159.

    Article  MathSciNet  MATH  Google Scholar 

  253. Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston.

    MATH  Google Scholar 

  254. Mandelbrot, B. (1983). The Fractal Geometry of Nature. W. H. Freeman. San Francisco.

    MATH  Google Scholar 

  255. Martirosian, D. H. (1986). Translation invariant Gibbs states in the q-state Potts model. Communications in Mathematical Physics 105, 281–290.

    Article  MathSciNet  Google Scholar 

  256. Mauldin, R. D. and Williams, S. C. (1986). Random recursive constructions: asymptotic geometry and topological properties. Transactions of the American Mathematical Society 295, 325–346.

    Article  MathSciNet  MATH  Google Scholar 

  257. Meester, R. (1989). An algorithm for calculating critical probabilities and percolation functions in percolation models defined by rotations. Ergodic Theory and Dynamical Systems 8, 495–509.

    MathSciNet  MATH  Google Scholar 

  258. Meester, R. (1992). Connectivity in fractal percolation. Journal of Theoretical Probability 5, 775–789.

    Article  MathSciNet  MATH  Google Scholar 

  259. Meester, R. (1994). Uniqueness in percolation theory; a review. Statistica Neerlandica 48, 237–252.

    Article  MathSciNet  MATH  Google Scholar 

  260. Meester, R. (1995). Equality of critical densities in continuum percolation. Journal of Applied Probability 32, 90–104.

    Article  MathSciNet  MATH  Google Scholar 

  261. Meester, R. and Häggström, O. (1995). Asymptotic shapes in stationary first passage percolation. Annals of Probability 23, 1511–1522.

    Article  MathSciNet  MATH  Google Scholar 

  262. Meester, R. and Häggström, O. (1995). Nearest neighbour and hard sphere models in continuum percolation (to appear).

    Google Scholar 

  263. Meester, R. and Nowicki, T. (1989). Infinite clusters and critical values in two-dimensional circle percolation. Israel Journal of Mathematics 68, 63–81.

    Article  MathSciNet  MATH  Google Scholar 

  264. Meester, R., Penrose, M., and Sarkar, A. (1995). The random connection model in high dimensions (to appear).

    Google Scholar 

  265. Meester, R., Roy, R., and Sarkar, A. (1994). Non-universality and continuity of the critical covered volume fraction in continuum percolation. Journal of Statistical Physics 75, 123–134.

    Article  MathSciNet  MATH  Google Scholar 

  266. Meester, R. and Steif, J. (1996). On the continuity of the critical value for long range percolation in the exponential case. Communications in Mathematical Physics 180, 483–504.

    Article  MathSciNet  MATH  Google Scholar 

  267. Meester, R. and Steif, J. (1995). Consistent estimation of percolation quantities (to appear).

    Google Scholar 

  268. Menshikov, M. V. (1986). Coincidence of critical points in percolation problems. Soviet Mathematics Doklady 33, 856–859.

    MathSciNet  MATH  Google Scholar 

  269. Menshikov, M. V. (1987). Quantitative estimates and rigorous inequalities for critical points of a graph and its subgraphs. Theory of Probability and its Applications 32, 544–547.

    Article  Google Scholar 

  270. Menshikov, M. V. (1987). Strict inequalities for critical probabilities of a regular graph and its subgraphs. Uspekhy Math. Nauk 42, 239–240.

    MathSciNet  Google Scholar 

  271. Menshikov, M. V., Molchanov, S. A., and Sidorenko, A. F. (1986). Percolation theory and some applications. Itogi Nauki i Techniki (Series of Probability Theory, Mathematical Statistics, Theoretical Cybernetics) 24, 53–110.

    MathSciNet  MATH  Google Scholar 

  272. Menshikov, M. V. and Pelikh, K. D. (1989). Percolation with several types of defects. Estimates of the critical probability on the square lattice, 778–785 (in translation). Mathematical Notes 46, 38–47 (in Russian).

    MathSciNet  Google Scholar 

  273. Menshikov, M. V. and Pelikh, K. D. (1993). Percolation method in a system of numerous particles. Proceeding of Third Conference on Probability Methods in Discrete Mathematics, Petrozavodsk, TVP/VSP, pp. 329–336.

    Google Scholar 

  274. Menshikov, M. V. and Zuev, S. A. (1993). Percolation method in a system of numerous particles. Probabilistic Methods in Discrete Mathematics (V. F. Kolchin, ed.), TVP/VSP, pp. 329–336.

    Google Scholar 

  275. Menshikov, M. V. and Zuev, S. A. (1993). Models of p-percolation. Probabilistic Methods in Discrete Mathematics (V. F. Kolchin, ed.), TVP Science Publishers, pp. 337–347.

    Google Scholar 

  276. Morrow, G., Schinazi, R., and Zhang, Y. (1994). The critical contact process on a homogeneous tree. Journal of Applied Probability 31, 250–255.

    Article  MathSciNet  MATH  Google Scholar 

  277. Nash-Williams, C. St. J. A. (1959). Random walks and electric currents in networks. Proceedings of the Cambridge Philosophical Society 55, 181–194.

    Article  MathSciNet  MATH  Google Scholar 

  278. Newman, C. M. (1986). Some critical exponent inequalities for percolation. Journal of Statistical Physics 45, 359–368.

    Article  MathSciNet  Google Scholar 

  279. Newman, C. M. (1986). Percolation theory: a selective survey of rigorous results. Advances in Multiphase Flow and Related Problems (G. Papanicolaou, ed.), SIAM, Philadelphia, pp. 147–167.

    Google Scholar 

  280. Newman, C. M. (1987). Inequalities for γ and related critical exponents in short and long range percolation. Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten, ed.), Springer, Berlin, pp. 229–244.

    Chapter  Google Scholar 

  281. Newman, C. M. (1987). Another critical exponent inequality for percolation: β≥2/σ. Journal of Statistical Physics 47, 695–699.

    Article  MathSciNet  Google Scholar 

  282. Newman, C. M. (1989). The 1/r 2 phase transition: a review. Proceedings of the IXth International Congress on Mathematical Physics (B. Simon, A. Truman, and I. M. Davies, eds.), Adam Hilger, Bristol, pp. 377–379.

    Google Scholar 

  283. Newman, C. M. (1991). Ising models and dependent percolation. Topics in Statistical Dependence (H. W. Block, A. R. Sampson, and T. H. Savits, eds.), IMS Lecture Notes, Monograph Series, vol. 16, pp. 395–401.

    Google Scholar 

  284. Newman, C. M. (1991). Topics in percolation. Mathematics of Random Media (W. Kohler and B. White, eds.), vol. 27, American Mathematical Society, Providence, pp. 41–51.

    Google Scholar 

  285. Newman, C. M. (1994). Disordered Ising systems and random cluster representations. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 247–260.

    Chapter  Google Scholar 

  286. Newman, C. M. (1995). A surface view of first-passage percolation. Proceedings of The International Congress of Mathematicians (S. D. Chatterji, ed.), Birkhäuser, Boston, pp. 1017–1023.

    Chapter  Google Scholar 

  287. Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Annals of Probability 23, 977–1005.

    Article  MathSciNet  MATH  Google Scholar 

  288. Newman, C. M. and Schulman, L. S. (1981). Infinite clusters in percolation models. Journal of Statistical Physics 26, 613–628.

    Article  MathSciNet  MATH  Google Scholar 

  289. Newman, C. M. and Stein, D. L. (1990). Broken symmetry and domain structure in Ising-like systems. Physical Review Letters 65, 460–463.

    Article  MathSciNet  MATH  Google Scholar 

  290. Newman, C. M. and Stein, D. L. (1992). Multiple states and thermodynamic limits in short-ranged Ising spin glass models. Physical Review B 46, 973–982.

    Article  Google Scholar 

  291. Newman, C. M. and Stein, D. L. (1993). Chaotic size dependence in spin glasses. Cellular Automata and Cooperative Systems (N. Boccara, E. Goles, S. Martinez and P. Picco, eds.), Kluwer, Dordrecht, pp. 525–529.

    Chapter  Google Scholar 

  292. Newman, C. M. and Stein, D. L. (1994). Spin glass model with dimension-dependent ground state multiplicity. Physical Review Letters 72, 2286–2289.

    Article  Google Scholar 

  293. Newman, C. M. and Stein, D. L. (1995). Random walk in a strongly inhomogeneous environment and invasion percolation. Annales de l’Institut Henri Poincaré 31, 249–261.

    MathSciNet  MATH  Google Scholar 

  294. Newman, C. M. and Stein, D. L. (1995). Broken ergodicity and the geometry of rugged landscapes. Physical Review E 51, 5228–5238.

    Article  Google Scholar 

  295. Newman, C. M. and Stein, D. L. (1996). Non-mean-field behavior of realistic spin glasses. Physical Review Letters 76, 515–518.

    Article  Google Scholar 

  296. Newman, C. M. and Stein, D. L. (1996). Ground state structure in a highly disordered spin glass model. Journal of Statistical Physics 82, 1113–1132.

    Article  MathSciNet  MATH  Google Scholar 

  297. Newman, C. M. and Stein, D. L. (1995). Spatial inhomogeneity and thermodynamic chaos (to appear).

    Google Scholar 

  298. Newman, C. M. and Volchan, S. B. (1996). Persistent survival of one-dimensional contact processes in random environments. Annals of Probability 24, 411–421.

    Article  MathSciNet  MATH  Google Scholar 

  299. Newman, C. M. and Wu, C. C. (1990). Markov fields on branching planes. Probability Theory and Related Fields 85, 539–552.

    Article  MathSciNet  MATH  Google Scholar 

  300. Onsager, L. (1944). Crystal statistics, I. A two-dimensional model with an order-disorder transition. The Physical Review 65, 117–149.

    Article  MathSciNet  MATH  Google Scholar 

  301. Ornstein, L. S. and Zernike, F. (1915). Accidental deviations of density and opalescence at the critical point of a single substance. Koninklijke Akademie van Wetenschappen te Amsterdam, Section of Sciences 17, 793–806.

    Google Scholar 

  302. Orzechowski, M. E. (1996). On the phase transition to sheet percolation in random Cantor sets. Journal of Statistical Physics 82, 1081–1098.

    Article  MathSciNet  MATH  Google Scholar 

  303. Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Annals of Probability 19, 1559–1574.

    Article  MathSciNet  MATH  Google Scholar 

  304. Pemantle, R. (1992). The contact process on trees. Annals of Probability 20, 2089–2116.

    Article  MathSciNet  MATH  Google Scholar 

  305. Pemantle, R. (1995). Uniform random spanning trees. Topics in Contemporary Probability and its Applications (J. L. Snell, ed.), CRC Press, Boca Raton.

    Google Scholar 

  306. Pemantle, R. and Peres, Y. (1994). Planar first-passage percolation times are not tight. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 261–264.

    Chapter  Google Scholar 

  307. Pemantle, R. and Peres, Y. (1994). Domination between trees and application to an explosion problem. Annals of Probability 22, 180–194.

    Article  MathSciNet  MATH  Google Scholar 

  308. Pemantle, R. and Peres, Y. (1996). On which graphs are all random walks in random environments transient?. Random Discrete Structures (D. Aldous and R. Pemantle, eds.), IMA Vol. Math. Appl. 76, Springer, New York, pp. 207–211.

    Chapter  Google Scholar 

  309. Pemantle, R. and Peres, Y. (1996). Galton-Watson trees with the same mean have the same polar sets. Annals of Probability 23, 1102–1124.

    Article  MathSciNet  MATH  Google Scholar 

  310. Penrose, M. (1996). Continuum percolation and Euclidean minimal spanning trees in high dimensions. Annals of Applied Probability 6, 528–544.

    Article  MathSciNet  MATH  Google Scholar 

  311. Penrose, M. and Pisztora, A. (1996). Large deviations for discrete and continuous percolation. Advances in Applied Probability 28, 29–52.

    Article  MathSciNet  MATH  Google Scholar 

  312. Peres, Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Communications in Mathematical Physics 177, 417–434.

    Article  MathSciNet  MATH  Google Scholar 

  313. Peyrière, J. (1978). Mandelbrot random beadsets and birthprocesses with interaction, I.B.M. Research Report RC-7417.

    Google Scholar 

  314. Pirogov, S. A. and Sinai, Ya. G. (1975). Phase diagrams of classical lattice systems. Theoretical and Mathematical Physics 25, 1185–1192.

    Article  MathSciNet  Google Scholar 

  315. Pirogov, S. A. and Sinai, Ya. G. (1976). Phase diagrams of classical lattice systems, continuation. Theoretical and Mathematical Physics 26, 39–49.

    Article  MathSciNet  Google Scholar 

  316. Pisztora, A. (1996). Surface order large deviations for Ising, Potts and percolation models. Probability Theory and Related Fields 104, 427–466.

    Article  MathSciNet  MATH  Google Scholar 

  317. Pokorny, M., Newman, C. M., and Meiron, D. (1990). The trapping transition in dynamic (invasion) and static percolation. Journal of Physics A: Mathematical and General 23, 1431–1438.

    Article  MathSciNet  Google Scholar 

  318. Potts, R. B. (1952). Some generalized order-disorder transformations. Proceedings of the Cambridge Philosophical Society 48, 106–109.

    Article  MathSciNet  MATH  Google Scholar 

  319. Preston, C. J. (1974). Gibbs States on Countable Sets. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  320. Propp J. G. and Wilson, D. B. (1995). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms (to appear).

    Google Scholar 

  321. Quas, A. (1996). Some properties of Lorentz lattice gas models (to appear).

    Google Scholar 

  322. Reimer, D. (1995). Butterflies (to appear).

    Google Scholar 

  323. Roy, R. and Meester, R. (1994). Uniqueness of unbounded occupied and vacant components in Boolean models. Advances in Applied Probability 4, 933–951.

    Article  MathSciNet  MATH  Google Scholar 

  324. Roy, R. and Meester, R. (1996). Continuum Percolation. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  325. Ruijgrok, T. W. and Cohen, E. G. D. (1988). Deterministic lattice gas models. Physics Letters A 133, 415–418.

    Article  MathSciNet  Google Scholar 

  326. Russo, L. (1978). A note on percolation. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 43, 39–48.

    Article  MathSciNet  MATH  Google Scholar 

  327. Russo, L. (1981). On the critical percolation probabilities. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 56, 229–238.

    Article  MathSciNet  MATH  Google Scholar 

  328. Russo, L. (1982). An approximate zero-one law. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 61, 129–139.

    Article  MathSciNet  MATH  Google Scholar 

  329. Scheinerman, E. R. and Wierman, J. C. (1987). Infinite AB percolation clusters exist. Journal of Physics A: Mathematical and General 20, 1305–1307.

    Article  MathSciNet  Google Scholar 

  330. Schonmann, R. H. (1994). Theorems and conjectures on the droplet-driven relaxation of stochastic Ising models. Probability and Phase Transition (G. R. Grimmett, ed.), Kluwer, Dordrecht, pp. 265–310.

    Chapter  Google Scholar 

  331. Seymour, P. D. and Welsh, D. J. A. (1978). Percolation probabilities on the square lattice. Advances in Graph Theory (B. Bollobás, ed.), Annals of Discrete Mathematics 3, North-Holland, Amsterdam, pp. 227–245.

    Chapter  Google Scholar 

  332. Simon, B. (1980). Correlation inequalities and the decay of correlations in ferromagnets. Communications in Mathematical Physics 77, 111–126.

    Article  MathSciNet  Google Scholar 

  333. Sinai, Ya. G. (1982). Theory of Phase Transitions: Rigorous Results. Pergamon Press, Oxford.

    Google Scholar 

  334. Slade, G. (1991). The lace expansion and the upper critical dimension for percolation. Mathematics of Random Media (W. E. Kohler and B. S. White, eds.), Lectures in Applied Mathematics, vol. 27, American Mathematical Society, Providence, pp. 53–63.

    Google Scholar 

  335. Slade, G. (1995). Bounds on the self-avoiding walk connective constant, Special Issue: Proceedings of the Conference in Honor of Jean-Pierre Kahane, 1993. Journal of Fourier Analysis and its Applications, 525–533.

    Google Scholar 

  336. Stacey, A. M. (1996). The existence of an intermediate phase for the contact process on trees. Annals of Probability 24, 1711–1726.

    Article  MathSciNet  MATH  Google Scholar 

  337. Stauffer, D. (1981). Scaling properties of percolation clusters. Disordered Systems and Localization (C. Castellani, C. DiCastro, and L. Peliti, eds.), Springer, Berlin, pp. 9–25.

    Chapter  Google Scholar 

  338. Swendsen, R. H. and Wang, J. S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters 58, 86–88.

    Article  Google Scholar 

  339. Sykes, M. F. and Essam, J. W. (1964). Exact critical percolation probabilities for site and bond problems in two dimensions. Journal of Mathematical Physics 5, 1117–1127.

    Article  MathSciNet  Google Scholar 

  340. Tanemura, H. (1995). Critical behavior for a continuum percolation model (to appear).

    Google Scholar 

  341. Vahidi-Asl, M. Q. and Wierman, J. C. (1990). First-passage percolation on the Voronoi tesselation and Delaunay triangulation. Random Graphs ’87, Wiley, London, pp. 341–359.

    MATH  Google Scholar 

  342. Vahidi-Asl, M. Q. and Wierman, J. C. (1990). A shape result for first-passage percolation on the Voronoi tesselation and Delaunay triangulation. Random Graphs ’89, Wiley, London, pp. 247–262.

    MATH  Google Scholar 

  343. Vahidi-Asl, M. Q. and Wierman, J. C. (1993). Upper and lower bounds for the route length of first-passage percolation in Voronoi tesselations. Bulletin of the Iranian Mathematical Society 19, 15–28.

    MathSciNet  MATH  Google Scholar 

  344. Wang, F. and Cohen, E. G. D. (1995). Diffusion in Lorentz lattice gas cellular automata: the honeycomb and quasi-lattices compared with the square and triangular lattices. Journal of Statistical Physics 81, 467–495.

    Article  MATH  Google Scholar 

  345. Watts, G. M. T. (1996). A crossing probability for critical percolation in two dimensions. Journal of Physics A: Mathematical and General 29, L363–L368.

    Article  MathSciNet  MATH  Google Scholar 

  346. Welsh, D. J. A. (1993). Percolation in the random-cluster process. Journal of Physics A: Mathematical and General 26, 2471–2483.

    Article  MathSciNet  MATH  Google Scholar 

  347. Whittle, P. (1986). Systems in Stochastic Equilibrium. John Wiley, Chichester.

    MATH  Google Scholar 

  348. Whittle, P. (1994). Polymer models and generalised Potts-Kasteleyn models. Journal of Statistical Physics 75, 1063–1092.

    Article  MathSciNet  MATH  Google Scholar 

  349. Wierman, J. C. (1987). Directed site percolation and dual filling models. Annals of Discrete Mathematics 33, 339–352.

    MathSciNet  MATH  Google Scholar 

  350. Wierman, J. C. (1987). Duality for k-degree percolation on the square lattice. Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten, ed.), Springer, Berlin, pp. 311–323.

    Chapter  Google Scholar 

  351. Wierman, J. C. (1988). Bond percolation critical probability bounds derived by edge contraction. Journal of Physics A: Mathematical and General 21, 1487–1492.

    Article  MathSciNet  Google Scholar 

  352. Wierman, J. C. (1988). AB percolation on close-packed graphs. Journal of Physics A: Mathematical and General 21, 1939–1944.

    Article  MathSciNet  MATH  Google Scholar 

  353. Wierman, J. C. (1988). On AB percolation in bipartite graphs. Journal of Physics A: Mathematical and General 21 1945–1949.

    Article  MathSciNet  Google Scholar 

  354. Wierman, J. C. (1989). AB percolation: a brief survey. Combinatorics and Graph Theory, vol. 25, Banach Centre Publications, Warsaw, pp. 241–251.

    MATH  Google Scholar 

  355. Wierman, J. C. (1990). Bond percolation critical probability bounds for the Kagomé lattice by a substitution method. Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh, eds.), Clarendon Press, Oxford, pp. 349–360.

    Google Scholar 

  356. Wierman, J. C. (1992). Equality of the bond percolation critical exponents for two pairs of dual lattices. Combinatorics, Probability, Computing 1, 95–105.

    Article  MathSciNet  MATH  Google Scholar 

  357. Wierman, J. C. (1994). Equality of directional critical exponents in multiparameter percolation models. Journal of Physics A: Mathematical and General 27, 1851–1858.

    Article  MathSciNet  MATH  Google Scholar 

  358. Wierman, J. C. (1995). Substitution method critical probability bounds for the square lattice site percolation model. Combinatorics, Probability, Computing 4, 181–188.

    Article  MathSciNet  MATH  Google Scholar 

  359. Wilson, R. J. (1979). Introduction to Graph Theory. Longman, London.

    MATH  Google Scholar 

  360. Wu, F. Y. (1982). The Potts model. Reviews in Modern Physics 54, 235–268.

    Article  Google Scholar 

  361. Yang, W. and Zhang, Y. (1992). A note on differentiability of the cluster density for independent percolation in high dimensions. Journal of Statistical Physics 66, 1123–1138.

    Article  MathSciNet  MATH  Google Scholar 

  362. Zhang, Y. (1991). A power law for connectedness of some random graphs at the critical point. Random Structures and Algorithms 2, 101–119.

    Article  MathSciNet  MATH  Google Scholar 

  363. Zhang, Y. (1992). Failure of the power laws on some subgraphs of the Z 2 lattice. Journal of Physics A: Mathematical and General 25, 6617–6622.

    Article  MathSciNet  MATH  Google Scholar 

  364. Zhang, Y. (1993). A shape theorem for epidemics and forest fires with finite range interactions. Annals of Probability 21, 1755–1781.

    Article  MathSciNet  MATH  Google Scholar 

  365. Zhang, Y. (1994). A note on inhomogeneous percolation. Annals of Probability 22, 803–820.

    Article  MathSciNet  MATH  Google Scholar 

  366. Zhang, Y. (1994). Analyticity properties at the supercritical state, preprint.

    Google Scholar 

  367. Zhang, Y. (1995). The fractal volume of the two-dimensional invasion percolation cluster. Communications in Mathematical Physics 167, 237–254.

    Article  MathSciNet  MATH  Google Scholar 

  368. Zhang, Y. (1995). Supercritical behaviors in first-passage percolation. Stochastic Processes and their Applications 59, 251–266.

    Article  MathSciNet  MATH  Google Scholar 

  369. Zhang, Y. (1995). A limit theorem for matching random sequences allowing deletions. Annals of Applied Probability 5, 1236–1240.

    Article  MathSciNet  MATH  Google Scholar 

  370. Zhang, Y. (1996). The complete convergence theorem on trees. Annals of Probability 24, 1408–1443.

    Article  MathSciNet  MATH  Google Scholar 

  371. Zhang, Y. (1996). Continuity of percolation probability in ∞+1 dimensions. Journal of Applied Probability 33, 427–433.

    MathSciNet  MATH  Google Scholar 

  372. Zhang, Y. (1996). Two critical behaviors of first passage time, preprint.

    Google Scholar 

  373. Zhang, Y. (1996). Some power laws on two dimensional critical bond percolation, preprint.

    Google Scholar 

  374. Zhang, Y. (1996). Divergence of the bulk resistance at criticality in disordered media. Journal of Statistical Physics 84, 263–267.

    Article  MathSciNet  MATH  Google Scholar 

  375. Zhang, Y. and Zhang, Y. C. (1984). A limit theorem for N 0n/n in first passage percolation. Annals of Probability 12, 1068–1076.

    Article  MathSciNet  MATH  Google Scholar 

  376. Ziff, R. M., Kong, X. P., and Cohen, E. G. D. (1991). Lorentz lattice-gas and kinetic-walk model. Physical Review A 44, 2410–2428.

    Article  Google Scholar 

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    Grimmett, G. (1997). Percolation and disordered systems. In: Bernard, P. (eds) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092620

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