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

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Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory Roberto Fern´andez J¨urg Fr¨ohlich Alan D. Sokal February 8, 2012 Mafalda cartoon “No dejes para ma˜nana” c Quino (Joaqu´ınLavado). Reprinted with permission. 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 1950’s, non-Markovian random-walk models, such as the self-avoiding walk, were introduced into theoretical polymer physics, and grad- ually 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 ϕ4 quantum field theory in space-time dimension d ( ) 4, and the proof of mean-field critical behavior 4 ≥ for ϕ 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 I II Foreword — 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 Ambjørn, Carlos Arag˜ao de Carvalho, David Barsky, An- ton Bovier, Sergio Caracciolo, Bergfinnur Durhuus, Paul Federbush, Giovanni Felder, Ross Graham, Gerhard Hartsleben, Thordur J´onsson and Antti Kupi- ainen. This work, carried out between roughly 1979 and the present, was much 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 4 an analysis of ϕd theory in d 4 space-time dimensions, and work by Dvoret- sky, Erd¨os, 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 walk, provided much additional stimulation for the work described in this volume. Symanzik’s article was fundamental, because it showed that Euclidean ϕ4 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-Erd¨os-Kakutani, Erd¨os-Taylor, and Lawler were important, because they showed that Brownian paths do not intersect in dimension d 4 and ≥ 1 that (weak) self-avoidance is an irrelevant constraint in dimension d ( ) 4. Jim ≥ 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 Erd¨os, Taylor and Lawler at hand, one might be able to prove a no-interaction (“triviality”) 4 theorem for continuum ϕ theory in space-time of dimension d ( ) 4. It was d ≥ 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, Fr¨ohlich [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 1See also the important recent work of Brydges and Spencer [93], Slade [471, 473, 472] and Hara and Slade [297, 293, 294] on self-avoiding walks in dimension d > 4. Foreword III 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 4 ϕd field theories in 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. 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-Fr¨ohlich-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 ϕ4 lattice field theories and Ising spin systems. In particular, we give 4 a detailed explanation of the triviality theorems for ϕd 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 4 superrenormalizable ϕd (d < 4), based on random-walk representations and IV Foreword 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- 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.
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