Abstract
Origins of first passage percolation problems. In 1957, Broadbent and Hammersley gave a mathematical formulation of percolation theory. Since then much work has been done in this field and has now led to first-passage percolation problems. In the following two examples we contrast the early formulation with its more recent developments.
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References
Bigelow, C. G.: Bibliography on project planning and control by network analysis 1959–61. Op. Res. 10, 728 (1962).
DooB, J. L.: Stochastic Processes. New York: Wiley 1952.
Erdös, P.: Remark on my paper “On a theorem of Hsu and Robbins” Ann. Math. Statist. 21, 138 (1950).
Feller, W.: An introduction to probability theory and its applications. New York: Wiley 1957.
Frisch, H. L., and J. M. Hammersley: Percolation processes and related topics. J. Soc. Indust. Appl. Math. 11, 894 (1963).
Fulkerson, D. R.: Expected critical path lengths in PERT networks. Op. Res. 10, 808 (1962).
Hammersley, J. M.: Generalization of the fundamental theorem on subadditive functions. Proc. Cambridge Phil. Soc. 58, 235 (1962).
Hille, E.: Functional Analysis and Semigroups. Amer. Math. Soc. Colloq. Publ. 1957, 31.
Kochen, M., C. Abraham, and E. Wong: Adaptive man-machine concept-proces-sing. Air Force Cambridge Research Laboratories Report No. 397 (1962).
Malcolm, D. G., J. H. Roseboom, E. E. Clark, and W. Fazar: Application of a technique for research and development program evaluation. Op. Res. 7, 646 (1959).
Pollack, M.: Solutions of the kth best route through a network — a review. To appear in J.Math. Anal and Appl.
Pollack, M., and W. Wiebenson: Solutions of the shortest route problem — a review. Op. Res. 8, 224 (1960).
Smith, W. L.: Renewal theory and its ramifications. J. Roy. Statist. Soc. B. 20, 243 (1958).
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Hammersley, J.M., Welsh, D.J.A. (1965). First-Passage Percolation, Subadditive Processes, Stochastic Networks, and Generalized Renewal Theory. In: Neyman, J., Le Cam, L.M. (eds) Bernoulli 1713, Bayes 1763, Laplace 1813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-49749-0_7
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DOI: https://doi.org/10.1007/978-3-642-49749-0_7
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