标题： 时空渗流 显示英文标题

标题： Space-time percolation

摘要： 疾病传播的接触模型可以看作是在$\ZZ \times \RR$上的一个有向渗流模型，其中连续轴的方向是随时间增加的方向。 渗流的技术使得对接触模型在临界点及其附近的较为完整的分析成为可能。 当时间轴无方向时对应的过程是一个无向渗流模型，现在可以应用标准技术对其进行分析。 可以以类似的方式在$\ZZ \times \RR$上构造一个随机簇模型，并伴随有连续的伊辛模型和庞茨模型。 这些模型除了提供具有横向场的量子伊辛模型的路径积分表示外，本身也具有独立的兴趣。 这种表示可用于获得在$\ZZ$上的量子伊辛模型中有限自旋集合的纠缠度的界限，其中纠缠度通过约化密度矩阵的熵来测量。 量子伊辛模型的平均场版本产生了一个在$K_n \times \RR$上的随机簇模型，从而扩展了在完全图$K_n$上的埃德罗斯-雷尼随机图。 显示英文摘要

摘要： The contact model for the spread of disease may be viewed as a directed percolation model on $\ZZ \times \RR$ in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the time-axis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a random-cluster model on $\ZZ \times \RR$, with associated continuum Ising and Potts models. These models are of independent interest, in addition to providing a path-integral representation of the quantum Ising model with transverse field. This representation may be used to obtain a bound on the entanglement of a finite set of spins in the quantum Ising model on $\ZZ$, where this entanglement is measured via the entropy of the reduced density matrix. The mean-field version of the quantum Ising model gives rise to a random-cluster model on $K_n \times \RR$, thereby extending the Erdos-Renyi random graph on the complete graph $K_n$.