Zero-temperature Glauber dynamics on $${\mathbb{Z}^d}$$
Zero-temperature Glauber dynamics on $${\mathbb{Z}^d}$$
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$${mathbb{Z}^d}$$ 上的零温格劳伯动力学
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发表时间：
2011
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通讯作者：
R. Morris
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作者：
R. Morris
我们研究了零温Glauber动力学$${\mathbb{Z}^d}$$，这是一个动态版本的伊辛模型的铁磁性。自旋最初是根据密度为p的伯努利分布选择的，然后状态根据多数规则连续（随机）更新。这对应于铁磁系统在高温下有外场时的突然淬火，对应于零温度下没有外场时的突然淬火。定义$${p_c（\mathbb{Z}^d）}$$为p上的下确界，使得系统以概率1固定在' + '。这是一个民间传说的猜想，对于每$${2 \le d \in \mathbb{N}}$$，$${p_c（\mathbb{Z}^d）= 1/2}$$。我们证明了当d → ∞时，${p_c（\mathbb{Z}^d）\to 1/2}$。
We study zero-temperature Glauber dynamics on $${\mathbb{Z}^d}$$ , which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define $${p_c(\mathbb{Z}^d)}$$ to be the infimum over p such that the system fixates at ‘ + ’ with probability 1. It is a folklore conjecture that $${p_c(\mathbb{Z}^d) = 1/2}$$ for every $${2 \le d \in \mathbb{N}}$$ . We prove that $${p_c(\mathbb{Z}^d) \to 1/2}$$ as d → ∞.