Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature
Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature
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DOI：
10.1007/s002200200658
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发表时间：
2002-07
影响因子：
2.4
通讯作者：
L. Fontes;R. Schonmann;V. Sidoravicius
中科院分区：
文献类型：
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作者：
L. Fontes;R. Schonmann;V. Sidoravicius
研究了一类连续时间Markov过程，它描述了超立方格点d（d≥ 2）上的± 1自旋翻转动力学，其初始自旋组态根据自旋密度p + 1的Bernoulli乘积测度选择.在演化过程中，每个位置的自旋在ratec= 0，或0 < α≤ 1，或1时翻转，这分别取决于该位置最近邻的大多数自旋是否存在并与给定位置的自旋值一致，或不存在（存在联系），或存在并与给定位置的自旋值不一致。这些动态对应于各种随机伊辛模型在0温度下，最近的邻居之间的均匀铁磁相互作用的哈密顿量。在α= 1的情况下，动态也是阈值投票模型。我们表明，如果p是足够接近1，那么系统固定在这个意义上，几乎每一个实现的初始配置和动力学演化，每个网站翻转只有100多次，最终达到状态+1。此外，我们还证明了在这种情况下，给定自旋在π处处于状态-1的概率q（t）满足如下的界：对于任意π> 0，q（t）≤ exp（-t（1/d）-π），对于larget。Ind= 2，我们得到互补界：对于任意的<$0，q（t）≥ exp（−t（1/2）+<$），对于larget。
We study a class of continuous time Markov processes, which describes ± 1 spin flip dynamics on the hypercubic latticeℤd,d≥ 2, with initial spin configurations chosen according to the Bernoulli product measure with densitypof spins + 1. During the evolution the spin at each site flips at ratec= 0, or 0 < α≤ 1, or 1, depending on whether, respectively, a majority of spins of nearest neighbors to this site exists and agrees with the value of the spin at the given site, or does not exist (there is a tie), or exists and disagrees with the value of the spin at the given site. These dynamics correspond to various stochastic Ising models at 0 temperature, for the Hamiltonian with uniform ferromagnetic interaction between nearest neighbors. In case α= 1, the dynamics is also a threshold voter model. We show that ifpis sufficiently close to 1, then the system fixates in the sense that for almost every realization of the initial configuration and dynamical evolution, each site flips only finitely many times, reaching eventually the state + 1. Moreover, we show that in this case the probabilityq(t) that a given spin is in state − 1 at timetsatisfies the bound: for arbitrary ɛ > 0,q(t) ≤ exp(−t(1/d) −ɛ), for larget. Ind= 2 we obtain the complementary bound: for arbitrary ɛ > 0,q(t) ≥ exp(−t(1/2) +ɛ), for larget.