Geometric analysis of \(\phi^ 4\) fields and Ising models. I, II. (English) Zbl 0533.58034

This is the first part of a detailed proof of a result announced earlier [the author, Phys. Rev. Lett. 47, 1-4 (1981); Mathematical Problems in Theoretical Physics, Proc. IAMP Conf. Berlin, 1981, Lect. Notes Phys. 153, 37–46 (1982)], concerning the free character of Euclidean \(\phi^ 4\) field theory in \(d>4\) dimensions. The analysis is nonperturbative and exploits the representation of field variables (or of spins in the isomorphic Ising models) as source-sink creation operators in a system of random currents. The onset of long-range order is attributed to percolation in an ensemble of sourceless currents and the interaction in the \(\phi^ 4\) field (or the critical behavior of the Ising models) is directly related to the intersection properties of long current clusters. Insight into the critical nature of \(d=4\) is derived from an analogy (due to Kurt Symanzik) with intersection properties of Brownian paths. It is also shown that in high dimensions \(d>4\) the critical behavior of Ising models is in exact agreement with mean field theory, but for \(d=2\) this is not true and hyperscaling is ”universal”.
Part I of the paper deals with Ising systems. Part II deals with the \(\phi^ 4\) field theory. An Appendix discusses the geometric aspects of the criticality of \(d=4\). Part III, which will be published separately, deals with random-surface phenomena, roughening transitions, stochastic geometry of \(\mathbb Z(2)\) lattice gauge theory and frustration in spin glasses.
Reviewer: M.E.Mayer


58J65 Diffusion processes and stochastic analysis on manifolds
58J90 Applications of PDEs on manifolds
81T08 Constructive quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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