  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
\lambda>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 \lambda. 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.
