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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

##### MSC:
 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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