# Random walks as Euclidean field theory (EFT)

Chapter

## Abstract

In this chapter we study a classic problem of probability theory — the intersection properties of simple random walks — using a rigorous blend of perturbation theory and renormalization-group arguments. Aside from its intrinsic mathematical interest, this problem played a key role in motivating the field-theoretic developments described in the remainder of this book, culminating in the triviality theorem for

*φ*^{4}field theory and Ising models in dimension*d*> 4. The logic of the field-theoretic arguments (described in more detail in Chapter 6) is roughly the following:- 1)
An

*identity*is derived which represents a lattice field theory in terms of*interacting*(“*non-simple*”) random walks. - 2)
*Inequalities*are derived which bound certain connected correlation functions of the field theory (in particular, the dimensionless renormalized coupling constant*g*) in terms of the*intersection properties*of these “field-theoretic” random walks. - 3)
The intersection properties of

*simple*random walks are used as*intuition*to motivate*conjectures*for the intersection properties of “field-theoretic” random walks (These conjectures must, however, be proven by different methods.)

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1992