Summary: We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which allows only emptying sites. We prove that the threshold density \(\rho_{c}\) for convergence to a completely empty configuration is non trivial, \(0< \rho_{c}<1\), contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, \(\rho < \rho_{c}\), emptying always occurs exponentially fast and that \(\rho_{c}\) coincides with the critical density for two-dimensional oriented site percolation on \(\mathbb Z^{2}\). This is known to occur also for some cellular automata with oriented rules for which the transition is continuous in the value of the asymptotic density and the crossover length determining finite size effects diverges as a power law when the critical density is approached from below. Instead for our model we prove that the transition is discontinuous and at the same time the crossover length diverges faster than any power law. The proofs of the discontinuity and the lower bound on the crossover length use a conjecture on the critical behaviour for oriented percolation. The latter is supported by several numerical simulations and by analytical (though non rigorous) works through renormalization techniques.
Finally, we discuss why, due to the peculiar mixed critical/first order character of this transition, the model is particularly relevant to study glassy and jamming transitions. Indeed, we show that it leads to a dynamical glass transition for a kinetically constrained spin model.
Most of the results that we present are the rigorous proofs of physical arguments developed in a joint work with D. S. Fisher [Jamming percolation and glass transition in lattice models. Phys. Rev. Lett. 96, 035702 (2006), On the universality of jamming percolation: reply to the comment of Jeng and Schwarz. Phys. Rev. Lett. 98, 129602 (2007)].