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Mesoadditive processes and the specific conductivity of lattices

Published online by Cambridge University Press:  14 July 2016

Abstract

The hypercubic lattice with bonds of random electrical resistance affords a model for the specific conductivity of microscopically irregular material. The resulting stochastic process lies between a subadditive and a superadditive process: mesoadditive processes of this type provide several unsolved problems for pure mathematicians.

Type
Part 8 - Random Walks, Graphs and Networks
Copyright
Copyright © Applied Probability Trust 1988 

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References

[1] Akcoglu, M. A. and Krengel, U. (1984) Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323, 5367.Google Scholar
[2] Broadbent, S. R. and Hammersley, J. M. (1957) Percolation processes I. Crystals and mazes. Proc. Camb. Phil. Soc. 53, 629641.Google Scholar
[3] Burridge, R., Childress, S. and Papanicolaou, G., (Eds) (1982) Macroscopic Properties of Disordered Media., Lecture Notes in Physics 154, Springer-Verlag, Berlin.Google Scholar
[4] Chayes, J. T. and Chayes, L. (1986) Bulk transport properties and exponent inequalities for random resistor and flow networks. Commun. Math. Phys. 105, 133152.CrossRefGoogle Scholar
[5] Deutscher, G., Zallen, R. and Adler, J., (eds) (1983) Percolation structures and processes. Ann. Israel Phys. Soc. 5.Google Scholar
[6] Frisch, H. L. and Hammersley, J. M. (1963) Percolation processes and related topics. J. Soc. Indust. Appl. Math. 11, 894918.Google Scholar
[7] Golden, K. and Papanicolaou, G. (1983) Bounds for effective parameters of heterogeneous media by analytic continuation. Commun. Math. Phys. 90, 473491.Google Scholar
[8] Grimmett, G. and Kesten, H. (1984) First-passage percolation, network flows and electrical resistances. Z. Wahrscheinlichkeitsth. 66, 335366.Google Scholar
[9] Hammersley, J. M. (1959) Bornes supérieures de la probabilité critique dans un processus de filtration. Coloq. Intern. CNRS 87, 1737.Google Scholar
[10] Hammersley, J. M. (1970) A few seedlings of research. Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 345394.Google Scholar
[11] Hammersley, J. M. (1974) Postulates for subadditive processes. Ann. Prob. 2, 652680.CrossRefGoogle Scholar
[12] Hammersley, J. M. and Morton, K. W. (1954) Poor man's Monte Carlo. J.R. Statist. Soc. B 16, 2338.Google Scholar
[13] Hammersley, J. M. and Welsh, D. J. A. (1962) Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. Oxford (2) 13, 108110.Google Scholar
[14] Hammersley, J. M. and Welsh, D. J. A. (1965) First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. Bernoulli–Bayes–Laplace Anniversary Volume, 65110, Springer-Verlag, New York.Google Scholar
[15] Hammersley, J. M. and Whittington, S. G. (1985) Self-avoiding walks in wedges. J. Phys. A. Math. Gen. 18, 101111.Google Scholar
[16] Hammersley, J. M., Torrie, G. M. and Whittington, S. G. (1982) Self-avoiding walks interacting with a surface. J. Phys. A. Math. Gen. 15, 539571.Google Scholar
[17] Hille, E. (1948) Functional Analysis and Semigroups. Amer. Math. Soc. Colloq. Publ. 31.Google Scholar
[18] Hughes, B. D. and Ninham, B. W. (1983) The Mathematics and Physics of Disordered Media., Lecture Notes in Mathematics 1035, Springer-Verlag, Berlin.Google Scholar
[19] Jäger, W., Rost, H. and Tautu, P., (Eds) (1982) Biological Growth and Spread., Lecture Notes in Biomathematics 154, Springer-Verlag, Berlin.Google Scholar
[20] Kesten, H. (1963) On the number of self-avoiding walks. J. Math. Phys. 4, 960969; 5 (1964), 1128–1137.CrossRefGoogle Scholar
[21] Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhauser, Boston.CrossRefGoogle Scholar
[22] Kesten, H. (1986) Aspects of first-passage percolation. Ecole d'Eté de Probabilités de Saint-Flour XIV, ed. Hennequin, P., Lecture Notes in Mathematics 1180, Springer-Verlag, Berlin, 125264.Google Scholar
[23] Kesten, H. (1987) Percolation theory and first-passage percolation. Ann. Prob. 15, 12311271.CrossRefGoogle Scholar
[24] Kingman, J. F. C. (1973) The ergodic theory of subadditive stochastic processes. J.R. Statist. Soc. B 30, 499510.Google Scholar
[25] Kingman, J. F. C. (1973) Subadditive ergodic theory. Ann. Prob. 1, 883909.Google Scholar
[26] Kingman, J F. C. (1973) Subadditive processes. Ecole d'Eté de Probabilités de Saint-flour, Lecture Notes in Mathematics 539, Springer-Verlag, Berlin, 168223.Google Scholar
[27] Kirkpatrick, S. (1973) Percolation and conduction. Rev. Mod. Phys. 45, 574588.Google Scholar
[28] Krengel, U. (1985) Ergodic Theorems. de Gruyter, Berlin.CrossRefGoogle Scholar
[29] Künneman, R. (1983) The diffusion limit for reversible jump processes on Zd with ergodic random bond conductivities. Commun. Math. Phys. 90, 2768.CrossRefGoogle Scholar
[30] Logan, B. F. and Shepp, L. A. (1975) A variational problem for Young tableaux. Adv. Math. 26, 206222.Google Scholar
[31] Nguyen, X. X. (1980) Ergodic theorems for multidimensional subadditive processes. Adv. Appl. Prob. 12, 572573.Google Scholar
[32] Pólya, G. and Szëgo, G. (1925) Aufgaben und Lehrsätze aus der Analysis. Springer-Verlag, Berlin.Google Scholar
[33] Ruelle, D. (1969) Statistical Mechanics: Rigorous Results. Math. Phys. Monograph Series, Benjamin, Mass. Google Scholar
[34] Shante, V. K. S. and Kirkpatrick, S. (1971) An introduction to percolation theory. Adv. Phys. 20, 325357.Google Scholar
[35] Smythe, R. T. (1973) Strong laws of large numbers for r-dimensional arrays of strong variables. Ann. Prob. 1, 164170.CrossRefGoogle Scholar
[36] Smythe, R. T. (1976) Multiparameter subadditive processes. Ann. Prob. 4, 772782.Google Scholar
[37] 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
[38] Straley, J. P. (1977) Critical exponents for the conductivity of random resistor lattices. Phys. Rev. B 15, 57335737.Google Scholar
[39] Vershik, A. M. and Kerov, S. V. (1977) Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tables. Soviet Math. Dokl. 18, 527531.Google Scholar
[40] Ziman, J. (1979) Models of Disorder. Cambridge University Press.Google Scholar