Loss of tension in an infinite membrane with holes distributed by Poisson law

Abstract

If one randomly punches holes in an infinite tensed membrane, when does the tension cease to exist? This problem was introduced by R. Connelly in connection with applications of rigidity theory to natural sciences. We outline a mathematical theory of tension based on graph rigidity theory and introduce several probabilistic models for this problem. We show that if the ``centers'' of the holes are distributed in R^2 according to Poisson law with parameter λ>0, and the distribution of sizes of the holes is independent of the distribution of their centers, the tension vanishes on all of R^2 for any value of λ. In fact, it follows from a more general result on the behavior of iterative convex hulls of connected subsets of R^d, when the initial configuration of subsets is distributed according to Poisson law and the sizes of the elements of the original configuration are independent of this Poisson distribution. For the latter problem we establish the existence of a critical threshold in terms of the number of iterative convex hull operations required for covering all of R^d. The processes described in the paper are somewhat related to bootstrap and rigidity percolation models.