Percolación de lados sobre Zd y el Teorema de Harris-Kesten
Keywords:
Percolation model, exponential decay , uniqueness of the open cluster, FKG inequality , Russo’s formulaAbstract
These lecture notes are based on a mini-course “Percolation models” which I taught at National University ofEngineering in January 2014. The goal was to try to develop a first self-contained course in percolation models,that is one of the simplest models of statistical physics exhibiting a phase transition, and present some fundamentaltools that we use in the formulation and proof of Harris-Kesten Theorem (see [8] and [1]) on the exact value of thecritical probabilitypcfor bond percolation on Z2
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References
1. Kesten H., The critical probability of bond percolation on the square lattice equals 1
2 . Comm. Math. Phys. Vol. 74(1), Pag 41-59 (1980). 2. Kesten H., Percolation Theory for Mathematicians. Birkhauser(1982).
3. Grimmett, G. R.: Percolation. Springer-Verlag, Berlin (1999).
4. Bollobas, B., Riordan O.: Percolation. Cambridge University Press, United Kingdom (2006).
5. Aizenman, M., Kesten, H. , Newman, C.M. : Uniqueness of the innite cluster and continuity of connectivity functions for short and long range percolation. Communications in Mathematical Physics 111, 505-532 (1987)
6. Aizenman, M., Barsky D.: Sharpness of the phase transition in percolation models. Communications in Mathematical Physics 108, 489-526 (1987).
7. Menshikov M.V.: Coincidence of critical points in percolation problems. Soviet Mathematics Doklady 33, 856-859 (1986).
8. Harris T., A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc., 56, 13-20 (1960).
9. Broadbent S.R., Hammersley J.M.Percolation processes I. Cristals and mazes. Proc. Cambridge Philos. Soc., 53, 629 -641 (1957).
10. Hammersley J.M.Percolation processes: Lower bounds for the critical probability. Ann. Math. Statist, 28, 790-795 (1957).
11. van der Berg, J. , Maes, C.: Disagrement percolation in the study of Markov elds. Ann. Probab. 22, 749{763 (1994).
12. Krikun, M., Yambartsev, A.: Phase transition for the Ising model on the critical Lorentzian triangulation. Journal of Statistical Physics, v. 148, p. 422-439 (2012).
13. Haggstrom, O.: Markov random elds and percolation on general graphs. Adv. in Appl. Probab. v. 32, p. 39-66 (2000).
14. Grimmett, G. R.: Space-Time Percolation. Progress in Probability. v. 60, p. 305{320 (2008).
15. Fontes L.R.:Notas en percolac~ao. IMPA (1998).
16. Grimmett, G. R.: The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab.. v. 23(4), p. 1461{1510 (1995).
17. Georgii, H., Haggstrom, O. : The random geometry of equilibrium phases. Phase transitions and critical phenomena, Vol 18. Academic press, San Diego, CA , 1{142 (2001).
18. Fortuin, C.M., Kasteleyn, R.W.: On the random-cluster model I. Introduction and relation to other models. Physica, 57, 536{564 (1972).
19. Fortuin, C. M., Kasteleyn, P. W., Ginibre, J.: Correlation inequalities on some partially ordered sets, Communications in Mathematical Physics 22, 89-103 (1971).
20. Holley, R.:Remarks on the FKG inequalities, Communications in Mathematical Physics 36, 227-231 (1974).
21. Russo, L.: On the critical percolation probabilities, Zeitschrift fur Wahrscheinlichkeitstheorie and Verwandte Gebiete 56, 229-237 (1981).
22. Burton, R.M., Keane M.: Density and uniqueness in percolation, Communications in Mathematical Physics 121, 501-505 (1989).
2 . Comm. Math. Phys. Vol. 74(1), Pag 41-59 (1980). 2. Kesten H., Percolation Theory for Mathematicians. Birkhauser(1982).
3. Grimmett, G. R.: Percolation. Springer-Verlag, Berlin (1999).
4. Bollobas, B., Riordan O.: Percolation. Cambridge University Press, United Kingdom (2006).
5. Aizenman, M., Kesten, H. , Newman, C.M. : Uniqueness of the innite cluster and continuity of connectivity functions for short and long range percolation. Communications in Mathematical Physics 111, 505-532 (1987)
6. Aizenman, M., Barsky D.: Sharpness of the phase transition in percolation models. Communications in Mathematical Physics 108, 489-526 (1987).
7. Menshikov M.V.: Coincidence of critical points in percolation problems. Soviet Mathematics Doklady 33, 856-859 (1986).
8. Harris T., A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc., 56, 13-20 (1960).
9. Broadbent S.R., Hammersley J.M.Percolation processes I. Cristals and mazes. Proc. Cambridge Philos. Soc., 53, 629 -641 (1957).
10. Hammersley J.M.Percolation processes: Lower bounds for the critical probability. Ann. Math. Statist, 28, 790-795 (1957).
11. van der Berg, J. , Maes, C.: Disagrement percolation in the study of Markov elds. Ann. Probab. 22, 749{763 (1994).
12. Krikun, M., Yambartsev, A.: Phase transition for the Ising model on the critical Lorentzian triangulation. Journal of Statistical Physics, v. 148, p. 422-439 (2012).
13. Haggstrom, O.: Markov random elds and percolation on general graphs. Adv. in Appl. Probab. v. 32, p. 39-66 (2000).
14. Grimmett, G. R.: Space-Time Percolation. Progress in Probability. v. 60, p. 305{320 (2008).
15. Fontes L.R.:Notas en percolac~ao. IMPA (1998).
16. Grimmett, G. R.: The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab.. v. 23(4), p. 1461{1510 (1995).
17. Georgii, H., Haggstrom, O. : The random geometry of equilibrium phases. Phase transitions and critical phenomena, Vol 18. Academic press, San Diego, CA , 1{142 (2001).
18. Fortuin, C.M., Kasteleyn, R.W.: On the random-cluster model I. Introduction and relation to other models. Physica, 57, 536{564 (1972).
19. Fortuin, C. M., Kasteleyn, P. W., Ginibre, J.: Correlation inequalities on some partially ordered sets, Communications in Mathematical Physics 22, 89-103 (1971).
20. Holley, R.:Remarks on the FKG inequalities, Communications in Mathematical Physics 36, 227-231 (1974).
21. Russo, L.: On the critical percolation probabilities, Zeitschrift fur Wahrscheinlichkeitstheorie and Verwandte Gebiete 56, 229-237 (1981).
22. Burton, R.M., Keane M.: Density and uniqueness in percolation, Communications in Mathematical Physics 121, 501-505 (1989).
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Published
2021-06-18
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Cerda Hernández, J. (2021). Percolación de lados sobre Zd y el Teorema de Harris-Kesten. REVCIUNI, 16(1), 7–13. Retrieved from https://www.revistas.uni.edu.pe/index.php/revciuni/article/view/937
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