Summary
An improvement of Harris' theorem on percolation is obtained; it implies relations between critical points of matching graphs of the type of the one stated by Essam and Sykes. As another consequence, it is proved that the percolation probability, as a function of the probability of occupation of a given site, is infinitely differentiable, except at most in the critical point.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Frisch, H.L., Hammersley, J.M.: J. Soc. Industr. Appl. Math. 11, 894 (1963)
Shante, V.K.S., Kirkpatrick, S.: Advances Phys. 20, 325 (1971)
Essam, J.W.: In Phase Transitions and Critical Phenomena, eds. C. Domb and M.S. Green. New York: Academic Press 1973
Coniglio, A., Nappi, C.R., Peruggi, F., Russo, L.: J. Physics A, II. Ser. Math. Gen., 10, 205 (1977)
Lebowitz, J., Penrose, O.: Cluster and percolation inequalities for lattice systems with interactions (preprint)
Harris, T.E.: Proc. Cambridge Philos. Soc. 56, 13 (1960)
Fisher, M.E.: J. Mathematical Phys. 2, 620 (1961)
Sykes, M.F, Essam, J.W.: J. Mathematical Phys. 5, 1117 (1964)
Miyamoto, M.: Comm. Math. Phys. 44, 169 (1975)
Coniglio, A., Nappi, C.R., Peruggi, F., Russo, L.: Comm. Math. Phys. 51, 315 (1976)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Russo, L. A note on percolation. Z. Wahrscheinlichkeitstheorie verw Gebiete 43, 39–48 (1978). https://doi.org/10.1007/BF00535274
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00535274