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

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.