Skip to main content Accessibility help
×
Home
Hostname: page-component-684bc48f8b-rk5l8 Total loading time: 1.382 Render date: 2021-04-14T03:41:34.639Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Mixed percolation on the square lattice

Published online by Cambridge University Press:  14 July 2016

John C. Wierman
Affiliation:
The Johns Hopkins University

Abstract

In a planar percolation model, faces of the underlying graph, as well as the sites and bonds, may be viewed as random elements. With this viewpoint, Whitney duality allows construction of a planar dual percolation model for each planar percolation model, which applies to mixed models with sites, bonds, and faces open or closed at random. Using self-duality for percolation models on the square lattice, information is obtained about the percolative region in the mixed model.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1984 

Access options

Get access to the full version of this content by using one of the access options below.

Footnotes

Research supported by the National Science Foundation under Grant No. MCS 78–01168 and MCS 81–18229.

References

Broadbent, S. R. and Hammersley, J. M. (1957) Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc. 53, 629641.CrossRefGoogle Scholar
Fisher, M. E. (1961) Critical probabilities for cluster size and percolation problems. J. Math. Phys. 2, 620627.CrossRefGoogle Scholar
Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89103.CrossRefGoogle Scholar
Frisch, H. L., Hammersley, J. M. and Welsh, D. J. A. (1962) Monte Carlo estimates of percolation probabilities for various lattices. Phys. Rev. 126, 949951.CrossRefGoogle Scholar
Hammersley, J. M. (1980) A generalization of McDiarmid's theorem for mixed Bernoulli percolation. Math. Proc. Camb. Phil. Soc. 88, 167170.CrossRefGoogle Scholar
Hammersley, J. M. and Welsh, D. J. A. (1980) Percolation theory and its ramifications. Contemporary Phys. 21, 593605.CrossRefGoogle Scholar
Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 1320.CrossRefGoogle Scholar
Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals . Comm. Math. Phys. 74, 4159.CrossRefGoogle Scholar
Kesten, H. (1981) Analyticity properties and power law estimates of functions in percolation theory. J. Statist. Phys. 25, 717756.CrossRefGoogle Scholar
Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhäuser, Boston.CrossRefGoogle Scholar
Russo, L. (1978) A note on percolation theory. Z. Wahrscheinlichkeitsth. 43, 3948.CrossRefGoogle Scholar
Seymour, P. D. and Welsh, D. J. A. (1978) Percolation probabilities on the square lattice. Ann. Discrete Math. 3, 227245.CrossRefGoogle Scholar
Shante, V. K. S. and Kirkpatrick, S. (1971) An introduction to percolation theory. Adv. Phys. 20, 325357.CrossRefGoogle Scholar
Smythe, R. T. and Wierman, J. C. (1978) First-passage Percolation on the Square Lattice. Springer Lecture Notes in Mathematics 671, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Sykes, M. F. and Essam, J. W. (1964) Exact critical percolation probabilities for site and bond problems in two dimensions. J. Math. Phys. 5, 11171127.CrossRefGoogle Scholar
Whitney, H. (1932) Nonseparable and planar graphs. Trans. Amer. Math. Soc. 34, 339362.CrossRefGoogle Scholar
Whitney, H. (1933) Planar graphs. Fundamenta Math. 21, 7384.Google Scholar
Wierman, J. C. (1978) On critical probabilities in percolation theory. J. Math. Phys. 19, 19791982.CrossRefGoogle Scholar
Wierman, J. C. (1981) Bond percolation on honeycomb and triangular lattices. Adv. Appl. Prob. 13, 298313.CrossRefGoogle Scholar
Wierman, J. C. (1982) Percolation theory. Ann. Prob. 10, 509524.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 13 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 14th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Mixed percolation on the square lattice
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Mixed percolation on the square lattice
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Mixed percolation on the square lattice
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *