摘要翻译：
我们引入了一类新的二维细胞自动机，它具有bootstrap类渗流动力学。每个位点可以是空的，也可以被单个粒子占据，动力学遵循一个确定性的更新规则，在离散时间只允许空位点。我们证明了收敛到完全空配置的阈值密度$\rho_c$是不平凡的，$0<\rho_c<1$,这与标准的bootstrap渗流相反。此外，我们还证明了在亚临界区$\Rho<\Rho_c$中，排空总是以指数速度发生，并且$\Rho_c$与$\Bz^2$上二维定向位渗流的临界密度一致。对于某些具有定向规则的元胞自动机来说，这也是已知的，对于这种自动机，在渐近密度和决定有限尺寸效应的交叉长度的值上的转变是连续的，当从下面接近临界密度时，它以幂律发散。相反，对于我们的模型，我们证明了过渡是{它不连续}，同时交叉长度发散{它比任何幂律都快}。不连续性和交叉长度下界的证明使用了定向渗流临界行为的猜想。后者得到了几个数值模拟的支持，并通过重整化技术得到了解析（尽管不严格）工作的支持。最后，我们将讨论为什么由于这种转变的特殊{It混合临界/一阶特性}，该模型特别适用于研究玻璃态和干扰态转变。事实上，我们将证明它导致了一个动力学约束自旋模型的动力学玻璃化转变。我们给出的大多数结果都是与D.S.Fisher共同工作的物理论据的严格证明。
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英文标题：
《Spiral Model: a cellular automaton with a discontinuous glass transition》
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作者：
Cristina Toninelli, Giulio Biroli
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最新提交年份：
2007
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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英文摘要：
  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 $\bZ^2$. 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.
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PDF链接：
https://arxiv.org/pdf/709.0378