We derive a precise Ornstein-Zernike asymptotic formula for the decay of the two-point function 〈Σ₀Σ_x〉_β in the general context of finite range Ising type models on ℤ^d. The proof relies in an essential way on the a-priori knowledge of the strict exponential decay of the two-point function and, by the sharp characterization of phase transition due to Aizenman, Barsky and Fernández, goes through in the whole of the high temperature region β<β_c. As a byproduct we obtain that for every β<β_c, the inverse correlation length ξ_β is an analytic and strictly convex function of direction.

Authors and Affiliations

1. Dipartimento di Matematica, Università di Bologna, piazza di Porta S. Donato 5, I-40126 Bologna, Italy. e-mail: campanin@dm.unibo.it, , , , , , IT

   Massimo Campanino

2. Faculty of Industrial Engineering, Technion, Haifa 3200, Israel. e-mail: ieioffe@ie.technion.ac.il, , , , , , IL

   Dmitry Ioffe

3. Laboratoire d'Analyse, Topologie et Probabilités, UMR-CNRS 6632, CMI, Université de Provence, 39 rue Joliot Curie, 13453 Marseille, France. e-mail: velenik@cmi.univ-mrs.fr, , , , , , FR

   Y van Velenik

Received: 10 January 2002 / Revised version: 19 June 2002 / Published online: 14 November 2002

Partly supported by Italian G. N. A. F. A, EC grant SC1-CT91-0695 and the University of Bologna. Funds for selected research topics.

Partly supported by the ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Science and Humanities.

Partly supported by the Swiss National Science Foundation grant #8220-056599.

Mathematics Subject Classification (2000): 60F15, 60K15, 60K35, 82B20, 37C30

Key words or phrases: Ising model – Ornstein-Zernike decay of correlations – Ruelle operator – Renormalization – Local limit theorems

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