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General first-passage percolation

Published online by Cambridge University Press:  01 July 2016

Colin McDiarmid*
Affiliation:
Wolfson College, Oxford
*
Postal address: Institute of Economics and Statistics, St. Cross Building, Manor Road, Oxford OX1 3UL, U.K.

Abstract

We present some general results related to first-passage percolation. These results involve families of random variables arranged in independent subfamilies. Applications are given to the study of random networks and to first-passage percolation.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1983 

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References

[1] Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables, with applications. Ann. Math. Statist. 38, 14661474.Google Scholar
[2] Ford, L. R. and Fulkerson, D. R. (1962) Flows in Networks. Princeton University Press, Princeton, NJ.Google Scholar
[3] Hammersley, J. M. (1961) Comparision of atom and bond percolation processes. J. Math. Phys. 2, 728733.CrossRefGoogle Scholar
[4] Hammersley, J. M. (1980) A generalisation of McDiarmid's theorem for mixed Bernoulli percolation. Math. Proc. Camb. Phil. Soc. 88, 167170.Google Scholar
[5] Hammersley, J. M. and Walters, R. S. (1963) Percolation and fractional branching processes. J. SIAM 11, 831839.Google Scholar
[6] Hammersley, J. M. and Welsh, D. J. A. (1965) First passage percolation, subadditive processes, stochastic networks and generalised renewal theory. In Bernoulli-Bayes-Laplace Anniversary Volume, ed. LeCam, L. M. and Neyman, J., Springer-Verlag, Berlin, 61110.Google Scholar
[7] Jogdeo, K. (1977) Association and probability inequalities. Ann. Statist. 5, 495504.Google Scholar
[8] Lehmann, E. L. (1966) Some concepts of dependence. Ann. Math. Statist. 37, 11371153.Google Scholar
[9] McDiarmid, C. J. H. (1980) Clutter percolation and random graphs. Math. Progr. Study 13, 1725.Google Scholar
[10] McDiarmid, C. J. H. (1981) General percolation and random graphs. Adv. Appl. Prob. 13, 4060.Google Scholar
[11] Oxley, J. G. and Welsh, D. J. A. (1979) On some percolation results of J. M. Hammersley. J. Appl. Prob. 16, 526540.Google Scholar
[12] Rüschendorf, L. (1982) Comparison of percolation probabilities. J. Appl. Prob. 19, 864868.Google Scholar
[13] Smythe, R. T. and Wierman, J. C. (1978) First Passage Percolation on the Square Lattice. Lecture Notes in Mathematics 671, Springer-Verlag, Berlin.Google Scholar
[14] Welsh, D. J. A. (1977) Percolation and related topics. Sci. Prog., Oxford 64, 6583.Google Scholar