Dependent random graphs and spatial epidemics

Annals of Applied Probability

Volumn 8, Issue 2, 1998, Pages 317-336

  Van Den Berg, J ^a   Grimmett, Geoffrey R ^b   Schinazi, Rinaldo B ^c  

^a CWI   (Netherlands)

^b Statistical Laboratory   (United Kingdom)

^c UNIVERSITY OF COLORADO   (United States)

Author keywords

Critical value; Dependent percolation; Interacting particle system; Percolation; Spatial epidemic

Indexed keywords

EID: 0032221185     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/aoap/1028903529     Document Type: Article

References (21)

1. AIZENMAN, M. and BARSKY, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489-526.
2. AIZENMAN, M. and GRIMMETT, G. (1991). Strict monotonicity for critical points in percolation and ferromagnetic models. J. Statist. Phys. 63 817-835.
3. ANDJEL, E. and SCHINAZI, R. (1996). A complete convergence theorem for an epidemic model. J. Appl. Probab. 33 741-748.
4. BERG, J. VAN DEN (1997). Some reflections on disjoint occurrences of events. Unpublished manuscript.
5. 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 Algorithms 8 199-212.
6. BERG, J. VAN DEN and KESTEN, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556-569.
7. BEZUIDENHOUT, C. E. and GRIMMETT, G. R. (1990). The critical contact process dies out. Ann. Probab. 18 1462-1482.
8. DURRETT, R. (1992). Multicolor particle systems with large threshold and range. J. Theoret. Probab. 5 127-152.
9. DURRETT, R. and NEUHAUSER, C. (1991). Epidemics with recovery in D = 2. Ann. Appl. Probab. 1 189-206.
10. DURRETT, R. and SCHINAZI, R. (1993). Asymptotic critical value for a competition model. Ann. Appl. Probab. 3 1047-1066.
11. GRIFFEATH, D. (1979). Additive and cancelative interacting particle systems. Lecture Notes in Math. 724. Springer, Berlin.
12. GRIMMETT, G. R. (1989). Percolation. Springer, Berlin.
13. GRIMMETT, G. R. (1996). Percolation and disordered systems. Ecole d'Eté de Probabilités de Saint Flour XXVI. Lecture Notes in Math. 153-300. Springer.
14. HARRIS, T. E. (1974). Contact interactions on a lattice. Ann. Probab. 2 969-988.
15. HARRIS, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 355-378.
16. KESTEN, H. (1981). Analyticity properties and power law estimates in percolation theory. J. Statist. Phys. 25 717-756.
17. KUULASMAA, K. (1982). The spatial general epidemic and locally dependent random graphs. J. Appl. Probab. 19 745-758.
18. LIGGETT, T. M. (1985). Interacting Particle Systems. Springer, New York.
19. MENSHIKOV, M. V. (1986). Coincidence of critical points in percolation problems. Soviet Math. Dokl. 33 856-859.
20. MENSHIKOV, M. V. and PELIKH, K. D. (1990). Percolation with several types of defects. Estimates of the critical probability on the square lattice. Math. Notes 46 38-47. (In Russian.) [English translation in Math. Notes Acad. Sci. USSR 46 778-785.]
21. REIMER, D. (1997). Proof of the van de Berg-Kesten inequality. (Revised version of 1994 paper entitled "Butterflies.") Unpublished manuscript.