Edge percolation on a random regular graph of low degree

Annals of Probability

Volumn 36, Issue 4, 2008, Pages 1359-1389

  Pittel, Boris ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Giant component; Percolation; Random graph; Threshold probability; Transition window

Indexed keywords

EID: 52049109174     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/07-AOP361     Document Type: Article

References (27)

1. ALDOUS, D. J. and PITTEL, B. (2000). On a random graph with immigrating vertices: Emergence of the giant component. Random Structures Algorithms 17 79-102. MR1774745
2. ALON, N., BENJAMÍNI, I. and STACEY, A. (2004). Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32 1727-1745. MR2073175
3. ARONSON, J., FRIEZE, A. and PITTEL, B. G. (1998). Maximum matchings in sparse random graphs: Karp-Sipser revisited. Random Structures Algorithms 12 111-177. MR1637403
4. BALOGH, J. and PITTEL, B. (2007). Bootstrap percolation on a random regular graph. Random Structures Algorithms 30 257-286. MR2283230
5. BENDER, E. A. and CANFIELD, E. R. (1978). The asymptotic number of labelled graphs with given degree sequences. J. Combin. Theory Ser. A 24 296-307. MR0505796
6. BENTAMINI, I. (2002). Private communication.
7. BOLLOBÁS, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311-316. MR0595929
8. BOLLOBÁS, B. (1982). The asymptotic number of unlabelled regular graphs. J. London Math. Soc. 89 201-206. MR0675164
9. BOLLOBÁS, B. (1984). The evolution of random graphs. Trans. Amer. Math. Soc. 286 257-274. MR0756039
10. BOLLOBÁS, B. (200.1). Random Graphs, 2nd ed. Cambridge Univ. Press. MR1864966
11. BORGS, C., CHAYES, J. T., VAN DER HOFSTAD, R., SLADE, G. and SPENCER, J. (2005). Random subgraphs of finite graphs. I. The scaling window under the triangle condition. Random Structures Algorithms 31137-184. MR2155704
12. BORGS, C., CHAYES, J. T., VAN DER HOFSTAD, R., SLADE, G. and SPENCER, J. (2005). Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab. 33 1886-1944. MR2165583
13. DURRETT, R. (1985). Some general results concerning the critical exponents of percolation processes. Z. Wahrsch. Verw. Gebiete 69 421-437. MR0787606
14. DURRETT, R. (2005). Probability: Theory and Examples, 3rd ed. Wadsworth and Brooks/Cole, Pacific Grove, CA. MR1068527
15. DURRETT, R. (2006). Random Graph Dynamics. Cambridge Univ. Press. MR2271734
16. ERDOS, P. and RÉNYI, A. (1960). On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5 17-61. MR0125031
17. GRIMMETT, G. (1989). Percolation. Springer, New York. MR0995460
18. JANSON, S., ŁUCZAK, T., KNUTH, D. E. and PITTEL, B. (1993). The birth of the giant component. Random Structures Algorithms 4 233-358. MR1220220
19. ŁUCZAK, T. (1990). Component behavior near the critical point of the random graph process. Random Structures Algorithms 1 287-310. MR1099794
20. ŁUCZAK, T. (1990). On the number of sparse connected graphs. Random Structures Algorithms 1171-174. MR1138422
21. ŁUCZAK, T., PITTEL, B. and WIERMAN, J. C. (1994). The structure of random graph near the point of the phase transition. Trans. Amer. Math. Soc. 341 721-748. MR1138950
22. MOLLOY, M. and REED, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6 161-179. MR1370952
23. MOLLOY, M. and REED, B. (1998). The size of the giant component of a random graph with a given degree sequence. Combin. Probab. Comput. 7 295-305. MR1664335
24. MOON, J. W. (1970). Counting Labelled Trees. Canadian Mathematics Congress, Montreal. MR0274333
25. PITTEL, B. (2001). On the largest component of the random graph at a near critical stage. J. Combin. Theory Ser. B 82 237-269. MR1842114
26. PITTEL, B., SPENCER, J. and WORMALD, N. (.1996). Sudden emergence of a giant k-core in a random graph. J. Combin. Theory Ser. B 67 111-151. MR1385386
27. WORMALD, N. C. (1995). Differential equations for random processes and random graphs. Ann. Appl. Probab. 5 1217-1235. MR1384372