Abstract
The resistivity of a one-dimensional lattice consisting of randomly distributed conducting and insulating sites is considered. Tunneling resistance of the form ρ0 ne bn is assumed for a cluster ofn adjacent insulating sites. In the thermodynamic limit, the mean resistance per site diverges at the critical filling fractionp c =e−b, while the mean square resistivity fluctuations diverge at the lower filling fraction\(p_{c_2 } \)=p 2c . Computer simulations of large but finite systems, however, show only a very weak divergence of resistivity atp c and no divergence of the fluctuations at\(p_{c_2 } \). For finite lattices, calculation of the resistivity at the critical filling is shown to be simply related to the Petersburg problem. Analytic expressions for the resistivity and resistivity fluctuations are obtained in agreement with the results of computer simulations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Derrida and H. Hilhorst,J. Phys. C 14:L539 (1981).
P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher,Phys. Rev. B 22:3519 (1980).
J. G. Simmons,J. Appl. Phys. 34:1793 (1963).
P. J. Reynolds, H. E. Stanley, and W. Klein,J. Phys. A 10:L203 (1977).
W. Feller, inAn Introduction to Probability Theory and Its Applications, 2nd ed. (Wiley, New York, 1957).
D. Bernoulli, inCommentarii Academiae Scientarium Imperialis Petropolitanae, Vol. 5 for years 1730–1731, p. 175, 1738.
I. Todhunter, inA History of Mathematical Theory of Probability (Chelsea, New York, 1949).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Palffy-Muhoray, P., Barrie, R., Bergersen, B. et al. Tunneling resistivity of a one-dimensional random lattice and the petersburg problem. J Stat Phys 35, 119–130 (1984). https://doi.org/10.1007/BF01017369
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01017369