Kuelske C.
Weakly Gibbsian representations for joint measures
of quenched lattice spin models
(294K, PS)
ABSTRACT. 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 the results of a previous paper [K6], 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.