Random threshold growth dynamics

Bohman, Tom; Gravner, Janko

doi:10.1002/(sici)1098-2418(199908)15:1<93::aid-rsa4>3.0.co;2-k

Cited by 8 publications

(7 citation statements)

References 14 publications

(27 reference statements)

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“…The observation of propagation fronts in a model of collective relaxation like the facilitated Ising model underscores the essential characteristic of this model of such relaxation and, presumably, the characteristic of the class of kinetics to which this model belongs. Perhaps the most useful description of this kinetics class is random threshold growth [35]. In 1991, Butler and Harrowell [11] pointed out that the relaxation in the 2D-2spin facilitated kinetic Ising model corresponded to growth from clusters of flippable spins, only a few of which were so configured so as to escape the ultimate trapping of the growth.…”

Section: Discussionmentioning

confidence: 99%

Macroscopic facilitation of glassy relaxation kinetics: Ultrastable glass films with frontlike thermal response

Léonard

Harrowell

2010

The Journal of Chemical Physics

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The recent experimental fabrication of ultrastable glass films, via vapor deposition [Swallen et al., Science 315, 353 (2007)] and the observation of frontlike response to the annealing of these films [Swallen et al., Phys. Rev. Lett. 102, 065503 (2009)] have raised important questions about the possibility of manipulating the properties of glass films and addressing fundamental questions about kinetics and thermodynamics of amorphous materials. Central to both of these issues is the need to establish the essential physics that governs the formation of the ultrastable films and their subsequent response. In this paper, we demonstrate that a generic model of glassy dynamics can account for the formation of ultrastable films, the associated enhancement of relaxation rates by a factor of 10(5), the observation of frontlike response to temperature jumps and the temperature dependence of the front velocity.

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Section: Discussionmentioning

confidence: 99%

Macroscopic facilitation of glassy relaxation kinetics: Ultrastable glass films with frontlike thermal response

Léonard

Harrowell

2010

The Journal of Chemical Physics

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“…Regarding the actual evolution of the cluster in time, a number of growth models have been suggested, for example, cellular automata [37,61] and their random equivalent, threshold growth automata [12,31], which have been used to model epidemics [2]. The latter includes the well-known Richardson model [59].…”

Section: 3mentioning

confidence: 99%

Detection of an anomalous cluster in a network

Arias-Castro¹,

Candès²,

Durand³

2011

Ann. Statist.

144

214

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We consider the problem of detecting whether or not, in a given sensor network, there is a cluster of sensors which exhibit an "unusual behavior." Formally, suppose we are given a set of nodes and attach a random variable to each node. We observe a realization of this process and want to decide between the following two hypotheses: under the null, the variables are i.i.d. standard normal; under the alternative, there is a cluster of variables that are i.i.d. normal with positive mean and unit variance, while the rest are i.i.d. standard normal. We also address surveillance settings where each sensor in the network collects information over time. The resulting model is similar, now with a time series attached to each node. We again observe the process over time and want to decide between the null, where all the variables are i.i.d. standard normal, and the alternative, where there is an emerging cluster of i.i.d. normal variables with positive mean and unit variance. The growth models used to represent the emerging cluster are quite general and, in particular, include cellular automata used in modeling epidemics. In both settings, we consider classes of clusters that are quite general, for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic, by far the most popular method in practice, achieves that same rate to within a logarithmic factor. Our results are not limited to the normal location model, but generalize to any one-parameter exponential family when the anomalous clusters are large enough.Comment: Published in at http://dx.doi.org/10.1214/10-AOS839 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…This property, which we call exact stability, is only valid under substantial assumptions, as the model has to have opposite structure, in an appropriate sense, from the additive one considered in [6]. In the process, we extend the result of [3] to obtain the Wulff characterization of the invariant shape. We also show that exact stability is far from rare; in fact, almost all members of arguably the most natural family of two-dimensional growth models, the threshold growth cellular automata with square neighborhood, are exactly stable.…”

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confidence: 99%

“…In fact, there are new phenomena, and the classification of Theorem 2 below becomes much more complex. Second, some of our techniques are intrinsically two-dimensional, such as the explicitly solvable example of Section 6, the lattice geometry and analytic number theory of Section 7, and even combinatorial properties studied in [3]. Nevertheless, some results-notably Theorem 1-do readily generalize to arbitrary dimension.…”

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confidence: 99%

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Random growth models with polygonal shapes

Gravner¹,

Griffeath²

2006

Ann. Probab.

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We consider discrete-time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half-space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases. In many cases, exact stability is observed. That is, for small perturbations, the shapes of the deterministic and random processes agree exactly. We give a complete characterization of such cases, and show that they are prevalent among threshold growth CA with box neighborhood. We also design a nontrivial family of CA in which the shape is exactly computable for all values of its probability parameter.Comment: Published at http://dx.doi.org/10.1214/009117905000000512 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Random threshold growth dynamics

Cited by 8 publications

References 14 publications

Macroscopic facilitation of glassy relaxation kinetics: Ultrastable glass films with frontlike thermal response

Macroscopic facilitation of glassy relaxation kinetics: Ultrastable glass films with frontlike thermal response

Detection of an anomalous cluster in a network

Random growth models with polygonal shapes

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