Space-time percolation

Geoffrey Grimmett

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arXiv · 0705.0506

Space-time percolation

Geoffrey Grimmett

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Abstract

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 $K_n$.

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BibTeX RIS

Geoffrey Grimmett. 2007-05-03. Space-time percolation. https://arxiv.org/abs/0705.0506

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