2007/05/03 by Geoffrey Grimmett, Grimmett, Geoffrey
Mathematics · #60K35 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60K35
paper · pdf · doi:10.48550/arxiv.0705.0506
arxiv created 2007/05/03 · arxiv updated 2009/12/01
The contact model for the spread of disease may be viewed as a directed percolation model on \ZZ × \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 × \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 Kn × \RR, thereby extending the Erdos-Renyi random graph on the complete graph Kn.