Percolation, path large deviations and weakly Gibbs states

We present a unified approach to establishing the Gibbsian character of a w ide class of non-Gibbsian states, arising in the Renormalisation Group theo ry. Inside the realm of the Pirogov-Sinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give ris e to weakly Gibbsian measures. In other words, we show that the Griffiths-P earce-Israel scenario of RG pathologies is carried by atypical configuratio ns. The renormalized measures are described by an effective interaction, wi th relative energies well-defined on a full measure set of configurations. In this way we complete the first part of the Dobrushin Restoration Program : to give a Gibbsian description to non-Gibbsian states. A disagreement per colation estimate is used in the proof to bound the decay of quenched corre lations through which the interaction potential is constructed. The percola tion is controlled via a novel type of pathwise large deviation theory.