Percolation of the loss of tension in an infinite triangular lattice

We introduce a new class of bootstrap percolation models where the local ru les are of a geometric nature as opposed to simple counts of standard boots trap percolation. Our geometric bootstrap percolation comes from rigidity t heory and convex geometry. We outline two percolation models: a Poisson mod el and a lattice model. Our Poisson model describes how defects-holes is on e of the possible interpretations of these defects-imposed on a tensed memb rane result in a redistribution or loss of tension in this membrane; the la ttice model is motivated by applications of Hooke spring networks to proble ms in material sciences. An analysis of the Poisson model is given by Mensh ikov et al.((4)) In the discrete set-up we consider regular and generic tri angular lattices on the plane where each bond is removed with probability l -p. The problem of the existence of tension on such lattice is solved by re ducing it to a bootstrap percolation model where the set of local rules fol lows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p < 1. Moreover, the complete rela xation of tension -as defined in Section 4-occurs in a finite time almost s urely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.