Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T02:01:50.806Z Has data issue: false hasContentIssue false
  • English
  • Français

The loss of tension in an infinite membrane with holes distributed according to a Poisson law

Part of: Discrete geometry General convexity Phase transformations in solids Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium) Equilibrium statistical mechanics

Published online by Cambridge University Press:  01 July 2016

M. V. Menshikov ,
K. A. Rybnikov  and
S. E. Volkov
M. V. Menshikov*
Affiliation:
University of Durham
K. A. Rybnikov*
Affiliation:
Cornell University
S. E. Volkov*
Affiliation:
University of Bristol
*
∗ Postal address: Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK.
∗∗ Postal address: Department of Mathematics, Cornell University, Ithaca, NY 14853, USA.
∗∗∗ Postal address: School of Mathematics, University of Bristol, Bristol BS8 1TW, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

What is the effect of punching holes at random in an infinite tensed membrane? When will the membrane still support tension? This problem was introduced by Connelly in connection with applications of rigidity theory to natural sciences. The answer clearly depends on the shapes and the distribution of the holes. We briefly outline a mathematical theory of tension based on graph rigidity theory and introduce a probabilistic model for this problem. We show that if the centers of the holes are distributed in ℝ2 according to a Poisson law with density λ > 0, and the shapes are i.i.d. and independent of the locations of their centers, the tension is lost on all of ℝ2 for any λ. After introducing a certain step-by-step dynamic for the loss of tension, we establish the existence of a nonrandom N = N(λ) such that N steps are almost surely enough for the loss of tension. Also, we prove that N(λ) > 2 almost surely for sufficiently small λ. The processes described in the paper are related to bootstrap and rigidity percolation.

Keywords

(Bootstrap) percolationPoisson processtensionrigidity

MSC classification

Primary: 52A22: Random convex sets and integral geometry 52C25: Rigidity and flexibility of structures 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B27: Critical phenomena
Secondary: 82C43: Time-dependent percolation 74N10: Displacive transformations
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 2 , June 2002 , pp. 292 - 312
Copyright
Copyright © Applied Probability Trust 2002 

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.)

References

Adler, J. (1991). Bootstrap percolation. Physica A 171, 453–470.Google Scholar
Aizenman, M. and Lebowitz, J. L. (1988). Metastability effects in bootstrap percolation. J. Phys. A 21, 3801–3813.Google Scholar
Aizenman, M., Kesten, H. and Newman, C. M. (1987). Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Commun. Math. Phys. 111, 505–531.CrossRefGoogle Scholar
Chalupa, J., Leath, P. L. and Reich, G. R. (1979). Bootstrap percolation on Bethe lattice. J. Phys. C. 12, L31–35.Google Scholar
Connelly, R. (1982). Rigidity and energy. Invent. Math. 66, 11–33.Google Scholar
Connelly, R. (1998). Holes in a membrane: tension percolation (talk, collaborators: J. Mitchell and K. Rybnikov). Geometry Section, CMS Winter Meeting, Kingston, 13–15 December 1998.Google Scholar
Connelly, R. and Whiteley, W. (1996). Second-order rigidity and prestress stability for tensegrity frameworks. SIAM J. Discrete Math. 9, 453–491.CrossRefGoogle Scholar
Crapo, H. and Whiteley, W. (1994). Spaces of stresses, projections and parallel drawings for spherical polyhedra. Beitr. Algebra Geom. 35, 259–281.Google Scholar
Dehghanpour, P. and Schonmann, R. H. (1997). A nucleation-and-growth model. Prob. Theory Relat. Fields 107, 123–135.Google Scholar
Duxbury, P. M., Jacobs, D. J., Thorpe, M. F. and Moukarzel, C. (1998). Floppy modes and the free energy: rigidity and connectivity percolation on Bethe lattices. Phys. Rev. E 59, 2084–2092.Google Scholar
Gravner, J. and McDonald, E. (1997). Bootstrap percolation in a polluted environment. J. Statist. Phys. 87, 915–927.CrossRefGoogle Scholar
Grimmett, G. (1999). Percolation. Springer, Berlin.Google Scholar
Holroyd, A. E. (1998). Existence and uniqueness of infinite component in generic rigidity percolation. Ann. Appl. Prob. 8, 944–973.Google Scholar
Holroyd, A. E. (2001). Rigidity percolation and boundary conditions. To appear in Ann. Appl. Prob. 11, 1063–1078.Google Scholar
Jacobs, D. and Thorpe, M. F. (1995). Generic rigidity percolation: the pebble game. Phys. Rev. Lett. 75, 4051–4054.Google Scholar
Jacobs, D. and Thorpe, M. F. (1996). Generic rigidity percolation in two dimensions. Phys. Rev. E 53, 3682–3693.Google Scholar
Kesten, H. (1982). Percolation Theory for Mathematicians (Progress Prob. Statist. 2). Birkhäuser, Boston.Google Scholar
Kogut, P. M. and Leath, P. L. (1981). Bootstrap percolation transitions on real lattices. J. Phys. C 14, 3187–3194.Google Scholar
Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Prob. 25, 71–95.Google Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
Menshikov, M. V. (1985). Estimates of percolation thresholds for lattices in Rd . Soviet Math. Dokl. 32, 368–370.Google Scholar
Menshikov, M. V. (1986). Methods of generations and projection. Estimates of critical probability for lattices in Rn in percolation problems. In Combinatorial Analysis, No. 7, ed. Rybnikov, K. A., Moskov. Gos. Univ., Moscow, pp. 5–13, 162 (in Russian).Google Scholar
Menshikov, M. V. and Sidorenko, A. F. (1987). The coincidence of critical points in Poisson percolation models. Theory Prob. Appl. 32, 547–550.Google Scholar
Menshikov, M. V., Molchanov, S. A. and Sidorenko, A. F. (1986). Percolation theory and some applications. Akad. Nauk. SSSR 24, 53–110 (in Russian). English translation: J. Soviet Math. 42, 1766–1810.Google Scholar
Rybnikov, K. (1999). Polyhedral partitions and stresses. , Queen's University, Kingston. Available at http://www.math.cornell.edu/~rybnikov/.Google Scholar
Schonmann, R. (1992). On the behavior of some cellular automata related to bootstrap percolation. Ann. Prob. 20, 174–193.Google Scholar
Tang, W. and Thorpe, M. F. (1987). Mapping between random central-force networks and random resistor networks. Phys. Rev. B 36, 3798–3804.Google Scholar
Tang, W. and Thorpe, M. F. (1988). Percolation of elastic networks under tension. Phys. Rev. B 37, 5539–5551.Google Scholar
Thorpe, M. F. (1983). Continuous deformations in random networks. J. Non-Cryst. Solids 57, 355–370.CrossRefGoogle Scholar
Thorpe, M. F. and Duxbury, P. M. (eds) (1999). Rigidity Theory and Applications. Kluwer/Plenum, New York.Google Scholar
Van Enter, A. C. D. (1987). Proof of Straley's argument for bootstrap percolation. J. Statist. Phys. 48, 943–945.Google Scholar
Zuev, S. A. (1987). Estimates for the percolation threshold for a square lattice. Theory Prob. Appl. 32, 606–609.Google Scholar
Zuev, S. A. and Sidorenko, A. F. (1985a). Continuous models of percolation theory. I. Teoret. Mat. Fiz. 62, 76–86 (in Russian).Google Scholar
Zuev, S. A. and Sidorenko, A. F. (1985b). Continuous models of percolation theory. II. Teoret. Mat. Fiz. 62, 253–262 (in Russian).Google Scholar