TY - JOUR

T1 - Percolation, path large deviations and weakly Gibbs states

AU - Maes, Christian

AU - Redig, Frank

AU - Shlosman, Senya

AU - Van Moffaert, Annelies

PY - 2000/2/1

Y1 - 2000/2/1

N2 - We present a unified approach to establishing the Gibbsian character of a wide class of non-Gibbsian states, arising in the Renormalisation Group theory. Inside the realm of the Pirogov-Sinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give rise to weakly Gibbsian measures. In other words, we show that the Griffiths-Pearce-Israel scenario of RG pathologies is carried by atypical configurations. The renormalized measures are described by an effective interaction, with 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 percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed. The percolation is controlled via a novel type of pathwise large deviation theory.

AB - We present a unified approach to establishing the Gibbsian character of a wide class of non-Gibbsian states, arising in the Renormalisation Group theory. Inside the realm of the Pirogov-Sinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give rise to weakly Gibbsian measures. In other words, we show that the Griffiths-Pearce-Israel scenario of RG pathologies is carried by atypical configurations. The renormalized measures are described by an effective interaction, with 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 percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed. The percolation is controlled via a novel type of pathwise large deviation theory.

UR - http://www.scopus.com/inward/record.url?scp=0034340702&partnerID=8YFLogxK

U2 - 10.1007/s002200050029

DO - 10.1007/s002200050029

M3 - Article

AN - SCOPUS:0034340702

VL - 209

SP - 517

EP - 545

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -