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22 September 2020
 
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A new class of cellular automata with a discontinuous transition
Cristina Toninelli ; Giulio Biroli ;
Date 15 Dec 2005
Subject Statistical Mechanics
AbstractWe introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be occupied by a single particle or empty 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. Although the dynamical rules do not break any lattice symmetry rho_c coincides with the critical density for two-dimensional oriented site percolation on 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 models 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 will 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 will 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 proof of physical arguments developed in a joint work with D.S.Fisher.
Source arXiv, cond-mat/0512335
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