Recent progress in rigorous percolation theory

Kesten, Harry

Colloque Paul Lévy sur les processus stochastiques (22-26 juin 1987. École Polytechnique, Palaiseau), Astérisque, no. 157-158 (1988), pp. 217-231.

[1] M. Aizenman and D. J. Barsky, Sharpness of the phase transition in percolation models, Comm. Math. Phys. 108 (1987), 489-526. | DOI | Zbl

See also D. J. Barsky's Ph.D. thesis, Rutgers University, 1987.

[2] M. Aizenman, H. Kesten and C. M. Newman, Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation, Comm. Math. Phys. (1987). | Zbl

[3] M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys. 36 (1984), 107-143. | DOI | Zbl

[4] M. Aizenman and C. M. Newman, Discontinuity of the percolation density in one-dimensional $1∕{|x-y|}^2$ percolation models, Comm. Math. Phys. 107 (1986), 611-648. | DOI | Zbl

[5] S. R. Broadbent and J. M. Hammersley, Percolation processes, Proc. Cambr. Phil. Soc. 53 (1957), 629-641 and 642-645. | DOI | Zbl

[6] J. T. Chayes and L. Chayes, Percolation and Random Media, in Critical Phenomena, Random Systems and Gauge Theories. Les Houches Session XLIII (1984), K. Osterwalder and R. Storm, eds., Elsevier Science Publishers, 1986. | Zbl

[7] J. T. Chayes and L. Chayes, The mean field bound for the order parameter of Bernoulli percolation, IMA Publ. in Math. and its Appl., vol. 8 (1987), 49-71, H. Kesten, ed., Springer-Verlag. | Zbl

[8] J. T. Chayes and L. Chayes, On the upper critical dimension of Bernoulli percolation, preprint, 1987. | Zbl

[9] J. T. Chayes, L. Chayes, G. Grimmett, H. Kesten and R. Schonmann, preprint (1987).

[10] M. Fisher, Critical probabilities for cluster size and percolation problems, J. Math. Phys. 2 (1961), 620-627. | DOI | Zbl

[11] A. Gandolfi, Uniqueness of the infinite cluster for stationary Gibbs states, to appear. | DOI | Zbl

[12] A. Gandolfi, G. Grimmett and L. Russo, On the uniqueness of the infinite cluster in the percolation model, to appear. | DOI | Zbl

[13] A. Gandolfi, M. Keane and L. Russo, On the uniqueness of the infinite occupied cluster in two-dimensional site percolation, to appear. | DOI | Zbl

[14] G. Grimmett, Percolation, to appear.

[15] J. M. Hammersley, Bornes supérieures de la probabilité critique dans un processus de filtration, pp. 17-37 in Le Calcul des Probabilités et ses Applications CNRS, Paris (1959). | Zbl

[16] T. E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambr. Phil. Soc. 56 (1960), 13-20. | DOI | Zbl

[17] H. Kesten, The critical probability of bond percolation on ${\mathbb{Z}}^2$ equals $\frac{1}{2}$, Comm. Math. Phys. 74 (1980), 41-59. | DOI | Zbl

[18] H. Kesten, Percolation Theory for Mathematicians, Birkhäuser, 1982. | DOI | Zbl

[19] H. Kesten, A scaling relation at criticality for 2D-percolation, IMA Publ. in Math. and its Appl., vol. 8 (1987), 203-212, H. Kesten, ed., Springer-Verlag. | Zbl

[20] H. Kesten, Scaling relations for 2D-percolation, Comm. Math. Phys. 109 (1987), 109-156. | DOI | Zbl

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

[22] B. G. Nguyen, Correlation lengths for percolation processes, J. Stat. Phys. 46 (1987), 517-523. | DOI | Zbl

[23] L. Russo, A note on percolation, Z. Wahrsch. verw. Geb. 43 (1978), 39-48. | DOI | Zbl

[24] L. Russo, On the critical percolation probabilities, Z. Wahrsch. verw. Geb. 56 (1981), 229-237. | DOI | Zbl

[25] P. D. Seymour and D. J. A. Welsh, Percolation probabilities on the square lattice, Ann. Discrete Math., 3 (1978), 227-245. | DOI | Zbl

[26] D. Stauffer, Scaling theory of percolation clusters, Phys. Reports 54, No. 1 (1979), 1-74. | DOI

[27] D. Stauffer, Scaling properties of percolation clusters, Lecture Notes in Physics, Vol. 149 (1981), 9-25, C. Castellani, C. D. Castro, L. Peliti, eds., Springer-Verlag.

[28] M. F. Sykes and J. W. Essam, Exact critical percolation probabilities for site and bond problems in two dimensions, J. Math. Phys. 5 (1964), 1117-1127. | DOI

[29] H. Tasaki, Hyperscaling inequalities for percolation, Preprint (1987). | DOI | Zbl

[30] B. Toth, A lower bound for the critical probability of the square lattice site percolation, Z. Wahrsch. verw. Geb. 69 (1985), 19-22. | DOI | Zbl

[31] J. Van Den Berg and M. Keane, On the continuity of the percolation probability function, Contemp. Math. 26 (1984), 61-65. | DOI | Zbl

[32] J. C. Wierman, Bond percolation on honeycomb and triangular lattices, Adv. Appl. Prob. 13 (1981), 293-313. | DOI | Zbl