Evidence for non-universal exponents in bootstrap percolation

Force-balance percolation

We study models of correlated percolation where there are constraints on the occupation of sites that mimic force balance, i.e., for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions. We prove rigorously that p(c) < 1 for the two-dimensional models studied. Numerical data indicate that the force-balance percolation transition is discontinuous with a growing crossover length, with perhaps the same form as the jamming percolation models, suggesting that all models belong to the same universality class with the same underlying mechanism driving the transition in both cases. We find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition. Finally, we study the dynamics of the culling procedure invoked to obtain the force-balance configurations and find a dynamical exponent similar to that found in sandpile models.

Geometry of dynamically available empty space is the key to near-arrest dynamics

We study several examples of kinetically constrained lattice models using dynamically accessible volume as an order parameter. Thereby we identify two distinct regimes exhibiting dynamical slowing, with a sharp threshold between them. These regimes are identified both by a new response function in dynamically available volume, as well as directly in the dynamics. Results for the self-diffusion constant in terms of the connected hole density are presented, and some evidence is given for scaling in the limit of dynamical arrest.

Directed rigidity and bootstrap percolation in 1+1 dimensions

We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. 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 transition belongs to the same universality class as the directed percolation. The same conclusion is reached by studying its surface critical behavior, i.e., the spreading of rigidity from finite clusters close to a nonrigid wall. Near 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 analytically. Our results are confirmed by numerical simulation.