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In this talk I will show that a large class of Gaussian Random Fields in the continuum or the lattice can be written as a sum of independent Gaussian random fields called fluctuation fields which enjoy the following properties: their covariances have finite range (compact support) and the fields are almost surely smooth. The fluctuation covariances satisfy very strong uniform bounds . After suitable rescaling the sequence of fluctuation fields converges in distribution to a a smooth continuum Gaussian random field whose covariance has finite range. This finite range multiscale expansion is the basis of a new mathematical form of Wilson's Renormalization Group where non local effects are minimized and estimates rendered simpler. In particular, on the lattice, this gives an alternative to the Kadanoff-Wilson renormalization group based on the block spin transformation. The talk is based on my joint work with D. Brydges and G. Guadagni (J. Stat.Phys. 115,415-449 (2004)) and a further paper with D. Brydges (2011, in preparation).

Commentaires: International Workshop at the Erwin Schroedinger International Institute for Mathematical Physics, University of Vienna.