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Spiral Model: a cellular automaton with a discontinuous glass transition | | Cristina Toninelli
; Giulio Biroli
; | | Date: |
4 Sep 2007 | | Abstract: | 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 $
ho_c$ for convergence to a completely empty configuration is non
trivial, $0<
ho_c<1$, contrary to standard bootstrap percolation. Furthermore
we prove that in the subcritical regime, $
ho<
ho_c$, emptying always occurs
exponentially fast and that $
ho_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 model we prove that the
transition is {it discontinuous} and at the same time the crossover length
diverges {it 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
{it 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
proofs of physical arguments developed in a joint work with D.S.Fisher. | | Source: | arXiv, 0709.0378 | | Services: | Forum | Review | PDF | Favorites | |
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