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Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory

  • Book
  • © 1992

Overview

Authors:
  1. Roberto Fernández

    1. Institut für Theoretische Physik, ETH Hönggerberg, Zürich, Switzerland

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  2. Jürg Fröhlich

    1. Institut für Theoretische Physik, ETH Hönggerberg, Zürich, Switzerland

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  3. Alan D. Sokal

    1. Department of Physics, New York University, New York, USA

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Part of the book series: Theoretical and Mathematical Physics (TMP)

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Table of contents (15 chapters)

  1. Front Matter

    Pages I-XVII
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  2. Critical phenomena, quantum field theory, random walks and random surfaces: Some perspectives

    1. Front Matter

      Pages 1-1
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    2. General introduction

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 3-36

    3. Phase transitions and critical points in classical spin systems: A brief survey

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 37-52

    4. Scale transformations and scaling (continuum) limits in lattice spin systems

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 53-57

    5. Construction of scaling limits: the renormalization group

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 59-77

    6. Random walks as Euclidean field theory (EFT)

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 79-99

    7. EFT as a gas of random walks with hard-core interactions

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 101-115

    8. Random-surface models

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 117-178

  3. Random-walk models and random-walk representations of classical lattice spin systems

    1. Front Matter

      Pages 179-179
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    2. Introduction

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 181-187

    3. Random-walk models in the absence of magnetic field

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 189-203

    4. Random-walk models in the presence of a magnetic field

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 205-211

    5. Factorization and differentiation of the weights

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 213-227

    6. Correlation inequalities: A survey of results

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 229-272

  4. Consequences for critical phenomena and quantum field theory

    1. Front Matter

      Pages 273-273
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    2. Background material

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 275-295

    3. Inequalities for critical exponents

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 297-366

    4. Continuum Limits

      • Roberto Fernández, Jürg Fröhlich, Alan D. Sokal
      Pages 367-403

  5. Back Matter

    Pages 405-444
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Keywords

About this book

Simple random walks - or equivalently, sums of independent random vari­ ables - have long been a standard topic of probability theory and mathemat­ ical physics. In the 1950s, non-Markovian random-walk models, such as the self-avoiding walk,were introduced into theoretical polymer physics, and gradu­ ally came to serve as a paradigm for the general theory of critical phenomena. In the past decade, random-walk expansions have evolved into an important tool for the rigorous analysis of critical phenomena in classical spin systems and of the continuum limit in quantum field theory. Among the results obtained by random-walk methods are the proof of triviality of the cp4 quantum field theo­ ryin space-time dimension d (::::) 4, and the proof of mean-field critical behavior for cp4 and Ising models in space dimension d (::::) 4. The principal goal of the present monograph is to present a detailed review of these developments. It is supplemented by a brief excursion to the theory of random surfaces and various applications thereof. This book has grown out of research carried out by the authors mainly from 1982 until the middle of 1985. Our original intention was to write a research paper. However, the writing of such a paper turned out to be a very slow process, partly because of our geographical separation, partly because each of us was involved in other projects that may have appeared more urgent.

Authors and Affiliations

  • Institut für Theoretische Physik, ETH Hönggerberg, Zürich, Switzerland

    Roberto Fernández, Jürg Fröhlich

  • Department of Physics, New York University, New York, USA

    Alan D. Sokal

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