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

445 Pages·1992·9.173 MB·English

by  Dr. Roberto Fernández, Professor Jürg Fröhlich, Professor Alan D. Sokal (auth.)

Texts and
Monographs
in Physics
w.
BeiglbOck
R.P.Geroch
E.H.Lieb
T.Regge
W. Thirring
Series Editors
Roberto Fernandez
JUrg Frohl ich
AI an D. Sakal
Random Walks,
Critical Phenomena,
and Triviality
in Quantum Field Theory
With 26 Figures
Springer-Verlag Berlin Heidelberg GmbH
Dr. Roberto Fernandez
Professor Jiirg Frohlich
Institut flir Theoretische Physik, ETH Hiinggerberg
CH-8093 Zurich, Switzerland
Professor Alan D. Sokal
Department of Physics, New York University
4, Washington Place, New York, NY 10003, USA
Editors
Wolf Beiglbock Tullio Regge
Institut flir Angewandte Mathematik Instituto di Fisica Teorica
Universitat Heidelberg Universita di Torino, C. so M. d' Azeglio, 46
1m Neuenheimer Feld 294 1-10125 Torino, Italy
W-6900 Heidelberg 1, FRO
Walter Thirring
Robert P. Geroch
Institut flir Theoretische Physik
Enrico Fermi Institute der Universitat Wien
University of Chicago Boltzmanngasse 5
5640 Ellis Ave. A-I090 Wien, Austria
Chicago, IL 60637, USA
Elliott H. Lieb
Jadwin Hall
P. O. Box 708
Princeton University
Princeton, NJ 08544-0708, USA
ISBN 978-3-662-02868-1
Library of Congress Cataloging-in-Publication Data. Fernandez, R. (Roberto), 1951-. Random Walks, critical
phenomena, and triviality in quantum field theory / R. Fernandez, J. Frohlich, A. D. Sokal. p. cm. --(Texts and
monographs in physics). Includes bibliographical references and index. 1. Quantum
ISBN 978-3-662-02868-1 ISBN 978-3-662-02866-7 (eBook)
DOI 10.1007/978-3-662-02866-7
field theory. 2. Random walks (Mathematics). 3. Critical phenomena (Physics). I. Frohlich, J. (JUrg.), 1946-. II.
Sokal, A. D. (Alan D.), 1955-. III. Title. IV. Series. QCI74.45.F46 1991 530.1'43--dc20 91-40850
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Foreword
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. Meanwhile,
other people and we found refinements and extensions of our original results, so
that the original plan for our paper had to be revised. Moreover, a preliminary
draft of our paper grew longer and longer. It became clear that our project
- if ever completed - would not take the shape of a paper publishable in a
theoretical or mathematical physics journal. We therefore decided to aim for
a format that is more expository and longer than a research paper, but too
specialized to represent something like a textbook on equilibrium statistical
mechanics or quantum field theory.
This volume reviews a circle of results in the area of critical phenomena in
spin systems, lattice field theories and random-walk models which are, in their
majority, due to Michael Aizenman, David Brydges, Tom Spencer and ourselves.
Other people have also been involved in these collaborations, among whom one
should mention Jan Ambj0rn, Carlos Aragiio de Carvalho, David Barsky, An
ton Bovier, Sergio Caracciolo, Bergfinnur Durhuus, Paul Federbush, Giovanni
Felder, Ross Graham, Gerhard Hartsleben, Thordur Jonsson and Antti Kupi
ainen. This work, carried out between roughly 1979 and the present, was much
VI Foreword
inspired by Symanzik's deep article "Euclidean Quantum Field Theory", which
appeared in the proceedings of the 1968 Varenna school on "Local Quantum
Theory" [494]. Papers by Schrader [451, 454, 452] and Sokal [481] devoted to
an analysis of <Pd theory in d 2 4 space-time dimensions, and work by Dvoret
sky, Erdos, Kakutani, Taylor and Lawler [327, 158, 164, 350, 351, 353, 352] on
intersection properties of Brownian paths and simple random walks and on self
avoiding walks, provided much additional stimulation for the work described in
this volume.
Symanzik's article was fundamental, because it showed that Euclidean <p4
field theory can be represented as a gas of weakly self-avoiding random paths
and loops. This is the main approach explored in this book. The articles by
Dvoretsky-Erdos-Kakutani, Erdos-Taylor, and Lawler were important, because
they showed that Brownian paths do not intersect in dimension d 2 4 and
that (weak) self-avoidance is an irrelevant constraint in dimension d 4.1 Jim
(2)
Glimm and Tom Spencer contributed much to making these ideas and results
popular among mathematical physicists.
It appeared that, with the ideas of Symanzik and the results of Erdos, Taylor
and Lawler at hand, one might be able to prove a no-interaction ("triviality")
theorem for continuum <Pd theory in space-time of dimension d 4. It was
(2)
pointed out in Sokal's thesis [481] that, for d > 4, it would be enough to prove a
tree-diagram bound on the connected four-point function to conclude triviality.
The right version of such a bound was first proven by Aizenman [4, 5], using a
"random-current representation" to prove new correlation inequalities. Shortly
afterwards, Frohlich [213] proved a slightly different version of this bound, using
Symanzik's random-path representation which also yields suitable correlation
inequalities. Their proofs were based, in an essential way, on the intuition that
random walks in dimension d > 4 do not intersect.
These results triggered much activity in developing random-current and
random-walk representations into a systematic tool for proving new correlation
inequalities which yield rigorous qualitative and quantitative information on
critical phenomena in lattice field theories and lattice spin systems. The best re
sults emerged for Ising models (sharpness of the phase transition, behavior of the
magnetization, bounds on critical exponents) and for the superrenormalizable
<Pd field theoriesin d = 2 and 3 dimensions (simple new proofs of existence and
nontriviality in the single-phase region, positivity of the mass gap, and asymp
toticity of perturbation theory). These ideas and methods were later extended
to percolation theory and to the theory of polymer chains, branched polymers,
lattice animals and random surfaces. In comparison to renormalization-group
constructions, the random-walk and random-current methods are somewhat soft
and less quantitative. They have, however, the advantage of closely following
simple and appealing intuitions, being mathematically precise and at the same
time technically rather simple.
ISee also the important recent work of Brydges and Spencer [93], Slade [471,473, 472J and
Hara and Slade [297, 293, 294J on self-avoiding walks in dimension d > 4.
Foreword VII
This book has three parts of rather different flavor. Part I is an overview of
critical phenomena in lattice field theories, spin systems, random-walk models
and random-surface models. We show how field-theoretic techniques can be used
to investigate the critical properties of random-walk models, and how random
walk representations can be used to analyze the critical properties of lattice
field theories and spin systems. We review the basic concepts involved in the
construction of scaling limits and in the renormalization group. The emphasis
is on ideas and concepts, not on complete mathematical arguments (which can,
however, be found in literature quoted). Chapter 7, co-authored by Gerhard
Hartsleben, contains a fairly detailed exposition of random-surface theory and
its various applications in statistical physics, quantum field theory and two
dimensional quantum gravity.
In Part II we systematize three different random-walk representations:
the Brydges-Frohlich-Spencer (BFS) representation of lattice spin systems, the
Aizenman (ARW) representation of the Ising model, and polymer-chain mod
els generalizing the self-avoiding walk. We introduce a common framework and
show that all the relevant results follow from just two properties of the weights:
repulsiveness "on the average" between walks, and attractiveness (or noninter
action) between nonoverlapping walks. This provides a unified explanation for
most of the correlation inequalities obtained in Chapter 12. Unfortunately, Part
II is somewhat technical; it will be of interest mainly to readers with specific
research interests in the random-walk representation.
In Part III we present a fairly systematic survey of what can be proven with
random-walk methods about critical exponents and the scaling (continuum)
limit for r.p4 lattice field theories and Ising spin systems. In particular, we give
a detailed explanation of the triviality theorems for <p~ models in space-time
dimension d > 4 and d = 4. We have tried hard to make Part III accessible to
readers who may not have read Part II.
A detailed overview of Parts II and III can be found in Chapter 8. We
remark that Parts II and III contain some previously unpublished results.
Our book is certainly not complete or exhaustive. Many things are only
sketched, others are not treated at all. For example, the new construction of
superrenormalizable r.p~ (d < 4), based on random-walk representations and
correlation inequalities, is reviewed only briefly in this book (Section 6.3). Re
sults on percolation theory, the superrenormalizable Edwards model of polymer
chains, branched polymers, . .. are not included at all.
If one thinks of all the progress that the renormalization group has made in
recent years, or if one thinks of the revolution that has recently occurred in our
understanding of two-dimensional conformal field theory and hence of critical
phenomena in two-dimensional statistical systems, one might argue that the
time for a book like ours has passed. However, we feel that some of the ideas
and methods explained in this book are interesting enough, or at least pretty
enough, to justify our attempt to collect and preserve them in the present form.
All the basic ideas underlying the main results reviewed in this book express
simple and appealing (physical or mathematical) intuitions. This feature distin-
VIII Foreword
guishes this book from some of the recent scientific production in mathematical
and theoretical physics which is growing more and more abstract and math
ematically sophisticated. We have the modest hope that our book is at least
agreeable scientific entertainment.
This book would never have seen the light of the day without the work of
Michael Aizenman, David Brydges and Tom Spencer, with whom we have had
the pleasure of several collaborations and countless interesting and important
discussions. Chapter 7 is the result of a collaboration with Gerhard Hartsleben,
whose deep insight helped to clarify our understanding of random-matrix models
and non-perturbative quantum gravity. He contributed with enormous enthusi
asm and devoted his best effort to the preparation and writing of this chapter.
Finally, Aernout van Enter, Takashi Hara and Gordon Slade read and made
helpful comments on parts of the manuscript. We wish to express our sincere
thanks to all these people, and to the many colleagues and friends whose work
and ideas have helped to determine the shape of this book.
Princeton, New Brunswick, Austin, Roberto Fernandez
Roma, Ziirich and New York, Jiirg Frohlich
December 1981'" .Lg$ ~ lWt1 1991 Alan D. Sokal
© Quino (Joaquin Lavado). Reprinted with permission.
Contents
Part I Critical phenomena, quantum field theory, random walks
and random surfaces: Some perspectives
1. General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Phenomenology of phase transitions
and critical phenomena ................................. 3
1.2 Multicritical points .................................... 11
1.3 Spin systems, quantum field theory and random walks:
An overview .......................................... 16
1.4 Classical lattice spin systems ............................ 17
1.5 Relativistic quantum field theory and Euclidean field theory 21
1.6 Some random-walk models .............................. 32
2. Phase transitions and critical points in classical spin systems:
A brief survey ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 Existence of phase transitions ........................... 37
2.2 Existence of critical points .............................. 47
2.3 Critical exponents in some exactly solvable models
and standard approximations ..... . . . . . . . . . . . . . . . . . . . . . . 47
2.4 Bounds on critical exponents ............................ 51
3. Scale transformations and scaling (continuum) limits
in lattice spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4. Construction of scaling limits: the renormalization group ........ 59
4.1 Block spin transformations ............................. 59
4.2 Fixed points of block spin transformations,
stable and unstable manifolds, critical exponents .......... 61
4.3 A brief sketch of the tree expansion ...................... 68
4.4 Rigorous uses of block spin transformations ............... 76
5. Random walks as Euclidean field theory (EFT) . . . . . . . . . . . . . . . . 79
5.1 Definition of the model and statement of results ........... 79
5.2 Heuristic renormalization-group argument ................ 81
5.3 Proof of lower bound ................................... 85
X Contents
5.4 Proof of upper bound .................................. 89
5.4.1 Expected number of intersections .................. 90
5.4.2 Rigorous renormilization-group argument ........... 91
5.4.3 Direct inclusion-exclusion argument .. . . . . . . . . . . . . . 94
6. EFT as a gas of random walks with hard-core interactions ....... 101
6.1 The Symanzik-BFS random-walk representation ........... 101
6.2 Consequences for triviality .............................. 106
6.3 Consequences for nontriviality ........................... 112
7. Random-surface models ..................................... 117
7.1 Continuum random-surface actions ...................... 118
7.2 Random-surface models in 7l.d ••••••••••••••••••••••••••• 121
7.2.1 Basic definitions ................................. 121
7.2.2 A mean-field theory for random-surface models ...... 124
7.2.3 The planar random-surface model ................. 128
7.3 Typical phenomena in random-surface theory ............. 132
7.4 Randomly triangulated random-surface models ............ 135
7.4.1 Definition of the model ........................... 135
7.4.2 Properties of the model .......................... 138
7.5 Random-matrix models ................................ 142
7.5.1 Matrix field theories ............................. 142
7.5.2 Random-matrix models and random triangulations:
Pure gravity .................................... 148
7.5.3 Random-matrix models for gravity coupled
to matter fields: Potts spins ..................... . 151
7.5.4 Random-matrix models for gravity coupled
to matter fields: Polyakov string ................... 153
7.6 The topological expansion: Non-perturbative results
for the pure gravity case ................................ 155
7.6.1 Simple vs. double scaling limit .................... 155
7.6.2 The mathematical toolbox ........................ 157
7.6.3 The simple scaling limit .......................... 163
7.6.4 The double scaling limit .......................... 165
7.6.5 Other approaches to the double scaling limit ........ 170
7.6.6 Perturbation of the string equation and KdV flow ... 174
7.6.7 Epilogue ....................................... 176
Part II Random-walk models and random-walk representations
of classical lattice spin systems
8. Introduction 181
9. Random-walk models in the absence of magnetic field . . . . . . . . . . 189
9.1 General definitions ..................................... 189
Contents XI
9.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.2.1 Polymer-chain models ............................ 192
9.2.2 BFS representation for continuous spin systems ..... 195
9.2.3 The ARW representation for the Ising model ........ 20l
10. Random-walk models in the presence of a magnetic field ........ 205
10.1 General definitions ..................................... 205
10.2 Examples ............................................. 207
10.2.1 Polymer-chain models ............................ 207
10.2.2 Baby polymer-chain models ....................... 208
10.2.3 BFS random-walk models ........................ 208
10.2.4 ARW model .................................... 210
11. Factorization and differentiation of the weights . . . . . . . . . . . . . . . . 213
11.1 Inequalities involving the partition of a family of walks ..... 213
11.1.1 Repulsiveness (or repulsiveness "on the average") .... 213
11.1.2 Attractiveness (or noninteraction)
between nonoverlapping (or compatible) walks ...... 218
11.2 Inequalities involving the splitting of a walk ............... 220
11.3 Differentiation of the weights with respect to J or h 223
11.3.1 Differentiation with respect to J .................. 223
11.3.2 Differentiation with respect to h ................... 226
12. Correlation inequalities: A survey of results . . . . . . . . . . . . . . . . . . . 229
12.1 Gaussian upper bounds ................................ 230
12.2 Truncated four-point function in zero magnetic field:
Lebowitz and Aizenman-Frohlich inequalities .............. 231
12.2.1 Upper bound (Lebowitz inequality) ................ 231
12.2.2 Nontrivial lower bounds
(Aizenman-Frohlich inequalities) .................. 232
12.2.3 Once-improved Aizenman-F'rohlich inequality
(Aizenman-Graham inequality) ................... 236
12.2.4 Twice-improved Aizenman-Frohlich inequality 240
12.3 Inequalities involving infinitely many orders
in the expansion parameter (non-Gaussian upper bounds) 241
12.3.1 General setup for the dilution trick ................ 242
12.3.2 Upper bound on the truncated four-point function
for the intersection properties
of ordinary random walks (IPORW model) ......... 246
12.3.3 Upper bound on the truncated four-point function
for the SAW .................................... 249
12.3.4 Upper bound on the truncated four-point function
for the ARW (Ising) model ....................... 252
12.4 The kernel K( x, y) and the truncated two-point function
in nonzero magnetic field ............................... 253
12.4.1 The kernel K(x, y) and the magnetization .......... 253