Skip to main content Accessibility help
×
Home
Hostname: page-component-6c8bd87754-4q2hw Total loading time: 0.189 Render date: 2022-01-19T05:49:15.211Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Percolation on an infinitely generated group

Published online by Cambridge University Press:  20 February 2020

Agelos Georgakopoulos*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
John Haslegrave
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
*
*Corresponding author.

Abstract

We give an example of a long range Bernoulli percolation process on a group non-quasi-isometric with ℤ, in which clusters are almost surely finite for all values of the parameter. This random graph admits diverse equivalent definitions, and we study their ramifications. We also study its expected size and point out certain phase transitions.

Type
Paper
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 639046).

References

Aizenman, M. and Newman, C. M. (1984) Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 107143.Google Scholar
Aizenman, M. and Newman, C. M. (1986) Discontinuity of the percolation density in one-dimensional 1/|xy|2 percolation models. Comm. Math. Phys. 107 611647.Google Scholar
Benjamini, I. and Schramm, O. (1996) Percolation beyond ℤd, many questions and a few answers. Electron. Comm. Probab. 1 7182.CrossRefGoogle Scholar
Benjamini, I. and Schramm, O. (2001) Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 23.Google Scholar
Duminil-Copin, H., Goswami, S., Raoufi, A., Severo, F. and Yadin, A. (2018) Existence of phase transition for percolation using the Gaussian free field. arXiv:1806.07733Google Scholar
Erdős, P. and Rényi, A. (1959) On random graphs I. Publ. Math. Debrecen 6 290297.Google Scholar
Georgakopoulos, A. (2016) Group-walk random graphs. In Groups, Graphs, and Random Walks (Ceccherini-Silberstein, T., Salvatori, M. and Sava-Huss, E., eds), Vol. 436 of the London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 190204.Google Scholar
Georgakopoulos, A. and Panagiotis, C. (2018) Analyticity results in Bernoulli percolation. arXiv:1811.07404Google Scholar
Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 1320.Google Scholar
van der Hofstad, R. and Nachmias, A. (2014) Unlacing hypercube percolation: A survey. Metrika 77 2350.CrossRefGoogle Scholar
van der Hofstad, R. and Nachmias, A. (2017) Hypercube percolation. J. Eur. Math. Soc. (JEMS) 19 725814.Google Scholar
Kaimanovich, V. A. and Vershik, A. M. (1983) Random walks on discrete groups: Boundary and entropy. Ann. Probab. 11 457490.Google Scholar
Kesten, H. (1981) Analyticity properties and power law estimates of functions in percolation theory. J. Statist. Phys. 25 717756.CrossRefGoogle Scholar
Lovász, L. and Szegedy, B. (2012) Random graphons and a weak Positivstellensatz for graphs. J. Graph Theory 70 214225.CrossRefGoogle Scholar
Newman, C. M. and Schulman, L. S. (1986) One dimensional 1/|j − i|s is percolation models: The existence of a transition for s ≤ 2. Comm. Math. Phys. 104 547571.Google Scholar
Penrose, M. (2003) Random Geometric Graphs, Vol. 5 of Oxford Studies in Probability, Oxford University Press.CrossRefGoogle Scholar
Schulman, L. S. (1983) Long range percolation in one dimension. J. Phys. A 16 L639L641.CrossRefGoogle Scholar
Steif, J. E. (2009) A Survey of Dynamical Percolation. In Fractal Geometry and Stochastics IV (Bandt, C., Zähle, M. and Mörters, P., eds), Vol. 61 of Progress in Probability, Birkhäuser, pp. 145174.Google Scholar
Woess, W. (2002) Random Walks on Infinite Graphs and Groups, Vol. 138 of Cambridge Tracts in Mathematics, Cambridge University Press.Google Scholar
2
Cited by

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.

Percolation on an infinitely generated group
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.

Percolation on an infinitely generated group
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.

Percolation on an infinitely generated group
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *