It may seem surprising to start our study with a description of Brownian motion. However, this offers an interesting introduction to the concept of Euclidean quantum field, and an intuitive understanding of the role of dimensionality. The effective (or Hausdorff) dimension two of Brownian curves is particularly significant. It means that two such curves fail to intersect, hence to interact, in dimension higher than four. This is illustrated in the first section of this chapter, which also discusses the transition from a discrete to a continuous walk. A similar analysis for interacting fields, pioneered by K. Symanzik, is presented in the second section. It is related to strong coupling, or high temperature, expansions, to be studied later, in particular in chapter 6 of this volume and chapter 7 of volume 2. The introduction of n-component fields provides the means to incorporate “self-avoiding” walks in the limit n → 0. We conclude this chapter with an analysis of elementary one-dimensional systems. This enables us to introduce the useful concept of transfer matrix.

Brownian motion

Random walks

We begin with a discussion of random walks on a regular, infinite lattice in d-dimensional Euclidean space. Each site has q neighbours, where q is called the coordination number of the lattice. At regular time intervals, separated by an amount Δt = 1, a walker jumps from one site towards a neighbouring one, chosen at random.