Directed rigidity and bootstrap percolation in 1+1 dimensions

We study directed rigidity percolation (equivalent to directed bootstrap pe rcolation) on three different lattices: square, triangular, and augmented t riangular. The first two of these display a first-order transition at p = 1 , while the augmented triangular lattice shows a continuous transition at a nontrivial p(c). On the augmented triangular lattice we find, by extensive numerical simulation, that the the directed rigidity percolation transitio n belongs to the same universality class as the directed percolation. The s ame conclusion is reached by studying its surface critical behavior, i.e., the spreading of rigidity from finite clusters close to a nonrigid wall. Ne ar the discontinuous transition at p = 1 on the triangular lattice, we are able to calculate the finite-size behavior of the density of rigid sites an alytically. Our results are confirmed by numerical simulation. [S1063-651X( 99)19210-9].