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Konstantin Rybnikov - Loss of tension in an infinite membrane with holes distributed by a Poisson law
| KONSTANTIN RYBNIKOV, Department of Mathematics and Statistics, Queen's University Kingston, Ontario K7L 3N6 |
| Loss of tension in an infinite membrane with holes distributed by a Poisson law |
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 show that if the ``centers'' of the holes are
distributed in
according to a Poisson law with
parameter
, and the distribution of the shapes of the
holes is independent of the distribution of their centers, the tension
vanishes on all of
for any value of 
. In fact,
this result follows from a more general theorem on the behavior of
iterative convex hulls of connected subsets of
, where
the initial configuration of subsets is distributed according to a
Poisson law, and the shapes 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
. The processes described in the paper
are somewhat related to bootstrap and rigidity percolation models.
This is a M. V. Menshikov and S. E. Volkov.
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