ar X iv : 0 70 5 . 05 06 v 1 [ m at h . PR ] 3 M ay 2 00 7 Space – time percolation

The contact model for the spread of disease may be viewed as a directed percolation model on Z×R 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 Z × R, 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 Z, 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 Kn × R, thereby extending the Erdős–Rényi random graph on the complete graph Kn.