Weakly Gibbsian representations for joint measures of quenched lattice spin models

Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spin-space and disorder-space be represented as (suitably generalized) Gibbs measures of an “annealed system”? - We prove that there is always a potential (depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure (“weak Gibbsianness”). This “positive” result is surprising when contrasted with earlier results, where we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”). In particular we gave natural “negative” examples where this set is even of measure one (including the random field Ising model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples. From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero. Our proof uses a martingale argument that allows to cut (an infinite-volume analogue of) the quenched free energy into local pieces, along with generalizations of Kozlov’s constructions.