Random walks, critical phenomena and triviality in quantum field theory.

*(English)*Zbl 0761.60061
Texts and Monographs in Physics. Berlin: Springer-Verlag. xiii, 444 p. (1992).

In the last decades various important results of the mathematical physics have been obtained by extensive applications of the techniques based on random walks. Notably the random walk methods provided the important tool for rigorous analysis of the critical phenomena in classical spin systems and the continuum limit in quantum field theory. Much credit for these developments should be given to the authors of this book so that the reader gets the first hand information and moreover a fair amount of original research material.

The fundamental phenomena explored in the book are the fact that Euclidean \(\varphi^ 4\) theory can be represented as a gas of weakly self-avoiding random paths and loops and that the Brownian paths do not intersect in dimension \(l\geq 4\). Deep analysis based on these ideas results in the proofs of new correlation inequalities which yield rigorous qualitative and quantitative information on critical phenomena in lattice field theories and lattice spin systems.

The book has three parts. In Part I the reader will find “an overview of critical phenomena in lattice field theories, spin systems, random-walk models and random-surface models”. In Part II the three different random walk representations are systematized. These are the Bridges-Frölich- Spencer representation of lattice spin systems, the Aizenman representation of the Ising models and polymer-chain models generalizing the self-avoiding walk. A common framework is introduced and it is shown that “all the relevant results follow from just two properties of the weights: repulsiveness on the average between walks, and attractiveness (or noninteraction) between nonoverlapping walks”. Part III contains a fairly systematic survey of what can be proven with random walk methods about critical exponents and the scaling (continuum) limit for \(\varphi^ 4_ d\) models in space time dimension \(d>4\) and \(d=4\).

The fundamental phenomena explored in the book are the fact that Euclidean \(\varphi^ 4\) theory can be represented as a gas of weakly self-avoiding random paths and loops and that the Brownian paths do not intersect in dimension \(l\geq 4\). Deep analysis based on these ideas results in the proofs of new correlation inequalities which yield rigorous qualitative and quantitative information on critical phenomena in lattice field theories and lattice spin systems.

The book has three parts. In Part I the reader will find “an overview of critical phenomena in lattice field theories, spin systems, random-walk models and random-surface models”. In Part II the three different random walk representations are systematized. These are the Bridges-Frölich- Spencer representation of lattice spin systems, the Aizenman representation of the Ising models and polymer-chain models generalizing the self-avoiding walk. A common framework is introduced and it is shown that “all the relevant results follow from just two properties of the weights: repulsiveness on the average between walks, and attractiveness (or noninteraction) between nonoverlapping walks”. Part III contains a fairly systematic survey of what can be proven with random walk methods about critical exponents and the scaling (continuum) limit for \(\varphi^ 4_ d\) models in space time dimension \(d>4\) and \(d=4\).

Reviewer: W.Karowski (Wrocław)

##### MSC:

60G50 | Sums of independent random variables; random walks |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

81T25 | Quantum field theory on lattices |

82B27 | Critical phenomena in equilibrium statistical mechanics |

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |