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

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

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: s.volkov@bristol.ac.uk
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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 

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