Percolación de lados sobre Zd y el Teorema de Harris-Kesten
Palabras clave:
Percolación, Decaimiento exponencial, Unicidad del cluster infinito, Desigualdad FKG, fórmula de RussoResumen
Estas notas son basadas en un minicurso "Modelos de percolación" que el autor dio en enero de 2014 en laUniversidad Nacional de Ingenier a. El objetivo fue introducir un primer curso autocontenido sobre modelos de per-colaci on, que es uno de los modelos mas simples de la fisica estadistica en presentar transicion de fase, y desarrollarlas herramientas necesarias para mostrar el celebrado resultado debido a Harris [8] y Kesten [1] sobre el c alculoexacto depcpara percolaci on de lados en Z2
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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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Publicado
2021-06-18
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Cerda Hernández, J. (2021). Percolación de lados sobre Zd y el Teorema de Harris-Kesten. Revista De La Facultad De Ciencias UNI, 16(1), 7–13. Recuperado a partir de https://revistas.uni.edu.pe/index.php/revciuni/article/view/937
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