The Hierarchical Model

We define a hierarchical Gaussian field in a way that is motivated by the finite-range decomposition of the Gaussian free field. The hierarchical Gaussian free field is a hierarchical field that has comparable large distance behaviour to the lattice Gaussian free field. We explicitly construct a version of the hierarchical Gaussian field and verify that it has the desired properties. We define the hierarchical |φ|⁴ model and state the main result proved in this book, which gives the critical behaviour of the susceptibility of the 4-dimensional hierarchical |φ|⁴ model. In preparation for the proof of the main result, we reformulate the hierarchical |φ|⁴ model as a perturbation of a Gaussian integral.