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Abstract

Random path representation and sharp correlations asymptotics at high-temperatures M. Campanino, D. Ioffe and Y. Velenik Stochastic Analysis on Large Scale Interacting Systems, Advanced Studies in Pure Mathematics 39 (T. Funaki and H. Osada ed.), 29-52 (2004). We recently introduced a robust approach to the derivation of sharp asymptotic formula for correlation functions of statistical mechanics models in the high-temperature regime. We describe its application to the nonperturbative proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the proof, in the Ising case, to arbitrary odd-odd correlation functions. We discuss the fluctuations of connection paths (invariance principle), and relate the variance of the limiting process to the geometry of the equidecay profiles. Finally, we explain the relation between these results from Statistical Mechanics and their counterparts in Quantum Field Theory. Key words: SAW, percolation, Ising model, Ornstein-Zernike decay of correlations, Ruelle operator, renormalization, local limit theorems. Files: PDF file, bibtex