Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T02:27:32.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations for discrete and continuous percolation

Part of: Equilibrium statistical mechanics Stochastic processes Special processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Mathew D. Penrose  and
Agoston Pisztora
Mathew D. Penrose*
Affiliation:
University of Durham
Agoston Pisztora
Affiliation:
Courant Institute, New York University
*
* Postal address: Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

Keywords

CONTINUUM PERCOLATIONLARGE DEVIATIONSSUPERCRITICAL PHASEPOISSON PROCESS

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60F10: Large deviations 82B43: Percolation
Secondary: 60G55: Point processes 82B21: Continuum models (systems of particles, etc.)
Type
Stochastic Geometry amd Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 1 , March 1996 , pp. 29 - 52
Copyright
Copyright © Applied Probability Trust 1996 

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

**

Present postal address: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA.

References

[1] Alexander, K. S., Chayes, J. T. and Chayes, L. (1990) The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Commun. Math. Phys. 131, 1–50.Google Scholar
[2] Chayes, J. T. and Chayes, L. (1986) Percolation and random media. In Les Houches 1984 Session XLIII: Critical Phenomena, Random Systems and Gauge Theoreies, Part II. ed. Osterwalder, K. and Stora, R. pp. 1001–1142. Elsevier, Amsterdam.Google Scholar
[3] Chayes, J. T., Chayes, L. and Newman, C. M. (1987) Bernoulli percolation above threshold; and invasion percolation analysis. Ann. Prob. 15, 1272–1287.Google Scholar
[4] Deuschel, J.-D. and Pisztora, A. (1995) Surface order large deviations for high-density percolation. Prob. Theory Rel. Fields. To appear.Google Scholar
[5] Deuschel, J.-D. and Stroock, D. (1989) Large Deviations. Academic Press, New York.Google Scholar
[6] Dunford, N. and Schwartz, J. T. (1958) Linear Operators. Vol. I. Wiley-Interscience, New York.Google Scholar
[7] Durrett, R. and Schonmann, R. H. (1988) Large deviations for the contact process and two dimensional percolation. Prob. Theory Rel. Fields 77, 583–603.CrossRefGoogle Scholar
[8] Gandolfi, A. (1989) Ph.D. Thesis. Technische Universiteit Delft.Google Scholar
[9] Grimmett, G. (1989) Percolation. Springer, New York.CrossRefGoogle Scholar
[10] Grimmett, G. and Marstrand, J. M. (1990) The supercritical phase of percolation is well-behaved. Proc. R. Soc. A 430, 439–457.Google Scholar
[11] Kesten, H. and Zhang, Y. (1990) The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Prob. 18, 537–555.Google Scholar
[12] Meester, R. and Roy, R. (1995) Continuum Percolation. Cambridge University Press, Cambridge.Google Scholar
[13] Penrose, M. D. (1991) On a continuum percolation model. Adv. Appl. Prob. 23, 536–556.CrossRefGoogle Scholar
[14] Penrose, M. D. (1995) Single linkage clustering and continuum percolation. J. Multivar. Anal. 53, 94–109.CrossRefGoogle Scholar
[15] Pisztora, A. (1993) Ph.D. Thesis. E.T.H., Zurich.Google Scholar
[16] Pisztora, A. (1994) Surface order large deviations for Ising, Potts and percolation models. Prob. Theory Rel. Fields. To appear.Google Scholar
[17] Roy, R. and Sarkar, A. (1992) Personal communication.Google Scholar
[18] Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, Chichester.Google Scholar
[19] Tanemura, H. (1993) Behavior of the supercritical phase of a continuum percolation model in d. J. Appl. Prob. 30, 382–396.Google Scholar