The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees

Annals of Applied Probability

Volumn 6, Issue 2, 1996, Pages 466-494

  Alexander, Kenneth S ^a  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

Central limit theorem; Continuum percolation; Minimal spanning tree; Occupied crossing

Indexed keywords

EID: 0030523782     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/aoap/1034968140     Document Type: Article

References (31)

1. ALDOUS, D. and STEELE, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247-258.
2. ALEXANDER, K. S. (1994). Rates of convergence of means for distance-minimizing subadditive Euclidean functionals. Ann. Appl. Probab. 4 902-922.
3. ALEXANDER, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87-104.
4. AVRAM, F. and BERTSIMAS, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 4 1033-1046.
5. BARBOUR, A. D., HOLST, L. and JANSON, S. (1992). Poisson Approximation. Clarendon, Oxford.
6. BEARDWOOD, J., HALTON, J. H. and HAMMERSLEY, J. M. (1959). The shortest path through many points. Proc. Cambridge Philos. Soc. 55 299-327.
7. CHAYES, J. T. and CHAYES, L. (1986). Percolation and random media. In Critical Phenomena, Random Systems and Gauge Theories (K. Osterwalder and R. Stora, eds.) 1001-1142. Elsevier, Amsterdam.
8. CHUNG, K. L. (1974). A Course in Probability Theory. Academic Press, New York.
9. EFRON, B. and STEIN, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586-596.
10. FEW, L. (1955). The shortest path and shortest road through n points. Mathematika 2 141-144.
11. GRIMMETT, G. R. (1989). Percolation. Springer, New York.
12. HARRIS, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13-20.
13. HIGUCHI, Y. (1993). Coexistence of infinite (*)-clusters. II. Ising percolation in two dimensions. Probab. Theory Related Fields 97 1-33.
14. JAILLET, P. (1992). Rates of convergence for quasi-additive smooth Euclidean functionals and application to combinatorial optimization problems. Math. Oper. Res. 17 964-980.
15. KESTEN, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.
16. KESTEN, H. and LEE, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495-527.
17. MEESTER, R. and ROY, R. (1996). Continuum Percolation. Cambridge Univ. Press.
18. PREPARATA, F. P. and SHAMOS, M. I. (1985). Computational Geometry: An Introduction. Springer, New York.
19. PRIM, R. C. (1957). Shortest connection networks and some generalizations. Bell System. Tech. J. 36 1389-1401.
20. RAMEY, D. B. (1982). A non-parametric test of bimodality with applications to cluster analysis. Ph.D. dissertation, Dept. Statistics, Yale Univ.
21. REDMOND, C. and YUKICH, J. (1994). Limit theorems and rates of convergence for Euclidean functionals. Ann. Appl. Probab. 4 1057-1073.
22. RHEE, W. T. (1994). Boundary effects in the traveling salesperson problem. Oper. Res. Lett. 16 19-25.
23. RHEE, W. T. and TALAGRAND, M. (1989). A sharp deviation inequality for the stochastic traveling salesman problem. Ann. Probab. 17 1-8.
24. 2. Ann. Probab. 18 1563-1575.
25. RUSSO, L. (1978). A note on percolation. Z. Wahrsch. Verw. Gebiete 43 39-48.
26. RUSSO, L. (1981). On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 229-237.
27. SEYMOUR, P. D. and WELSH, D. J. A. (1978). Percolation probabilities on the square lattice. In Advances in Graph Theory (B. Bollobas, ed.) 227-245. North-Holland, Amsterdam.
28. STEELE, J. M. (1981). Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann. Probab. 9 365-376.
29. STEELE, J. M. (1981). Complete convergence of short paths and Karp's algorithm for the TSP. Math. Oper. Res. 6 374-378.
30. STEELE, J. M. (1988). Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16 1767-1787.
31. ZUEV, S. A. and SIDORENKO, A. F. (1985). Continuous models of percolation theory. I, II. Theoret. and Math. Phys. 62 72-86, 253-265.