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We introduce a new class of two-dimensional cellular automata with abootstrap percolation-like dynamics. Each site can be either empty or occupiedby a single particle and the dynamics follows a deterministic updating rule atdiscrete times which allows only emptying sites. We prove that the thresholddensity $\rho_c$ for convergence to a completely empty configuration is nontrivial, $0<\rho_c<1$, contrary to standard bootstrap percolation. Furthermorewe prove that in the subcritical regime, $\rho<\rho_c$, emptying always occursexponentially fast and that $\rho_c$ coincides with the critical density fortwo-dimensional oriented site percolation on $\bZ^2$. This is known to occuralso for some cellular automata with oriented rules for which the transition iscontinuous in the value of the asymptotic density and the crossover lengthdetermining finite size effects diverges as a power law when the criticaldensity is approached from below. Instead for our model we prove that thetransition is {\it discontinuous} and at the same time the crossover lengthdiverges {\it faster than any power law}. The proofs of the discontinuity andthe lower bound on the crossover length use a conjecture on the criticalbehaviour for oriented percolation. The latter is supported by severalnumerical simulations and by analytical (though non rigorous) works throughrenormalization techniques. Finally, we will discuss why, due to the peculiar{\it mixed critical/first order character} of this transition, the model isparticularly relevant to study glassy and jamming transitions. Indeed, we willshow that it leads to a dynamical glass transition for a KineticallyConstrained Spin Model. Most of the results that we present are the rigorousproofs of physical arguments developed in a joint work with D.S.Fisher.

Spiral Model: a cellular automaton with a discontinuous glass transition