Random walks as Euclidean field theory (EFT)

  • Roberto Fernández
  • Jürg Fröhlich
  • Alan D. Sokal
Part of the Texts and Monographs in Physics book series (TMP)

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. 1)

    An identity is derived which represents a lattice field theory in terms of interacting (“non-simple”) random walks.

     
  2. 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. 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.)

     

Keywords

Summing 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Roberto Fernández
    • 1
  • Jürg Fröhlich
    • 1
  • Alan D. Sokal
    • 2
  1. 1.Institut für Theoretische PhysikETH HönggerbergZürichSwitzerland
  2. 2.Department of PhysicsNew York UniversityNew YorkUSA

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