Tension percolation
Robert Connelly, Cornell University

Abstract:  Consider a planar membrane in space clamped on its boundary with
inextendable internal material so that it supports tension.  What happens
when holes are created in this structure?  When will it still support
tension, and when will there be a floppy part that flexes in space?  We
present two classes of percolation models, both discrete, where tension can
exist in a natural sense, and where the creation of holes can have the
consequence of relieving the tension.  One approach is a continuous
bootstrap-like percolation of compact defects distributed with a Poisson
Law.  The other is bootstrap percolation on a triangular lattice.  In both
of these models it is the geometric properties of the underlying structure
(after the holes are created) that determines whether or not the tension
exists.  In many percolation problems there is a critical probability
parameter, where the property changes.  The answer for these models is
surprising.  This is joint work with K. Rybnikov and S. Volkov