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We study graphical representations for two related models. The first model isthe transverse field quantum Ising model, an extension of the original Isingmodel which was introduced by Lieb, Schultz and Mattis in the 1960's. Thesecond model is the space-time percolation process, which is closely related tothe contact model for the spread of disease. We consider a `space-time'random-cluster model and explore a range of useful probabilistic techniques forstudying it. The space-time Potts model emerges as a natural generalization ofthe quantum Ising model. The basic properties of the phase transitions in thesemodels are treated, such as the fact that there is at most one unboundedFK-cluster, and the resulting lower bound on the critical value in $\ZZ$. Wealso develop an alternative graphical representation of the quantum Isingmodel, called the random-parity representation. This representation is based onthe random-current representation of the classical Ising model, and allows usto study in much greater detail the phase transition and critical behaviour.The random-parity representation is used to prove sharpness of the phasetransition in the quantum Ising model, and to establish bounds on some criticalexponents. We address these issues by using the random-parity representation toestablish certain differential inequalities, integration of which give theresults. Finally we explore some consequences and possible extensions of theresults mentioned. For example, we determine the critical point for the quantumIsing model in $\ZZ$ and in `star-like' geometries.

Graphical representations of Ising and Potts models