20,445 research outputs found
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 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
How non-Gibbsianness helps a metastable Morita minimizer to provide a stable free energy
We analyze a simple approximation scheme based on the Morita-approach for the
example of the mean field random field Ising model where it is claimed to be
exact in some of the physics literature. We show that the approximation scheme
is flawed, but it provides a set of equations whose metastable solutions
surprisingly yield the correct solution of the model. We explain how the same
equations appear in a different way as rigorous consistency equations. We
clarify the relation between the validity of their solutions and the almost
surely discontinuous behavior of the single-site conditional probabilities.Comment: 15 page
Stability for a continuous SOS-interface model in a randomly perturbed periodic potential
We consider the Gibbs-measures of continuous-valued height configurations on
the -dimensional integer lattice in the presence a weakly disordered
potential. The potential is composed of Gaussians having random location and
random depth; it becomes periodic under shift of the interface perpendicular to
the base-plane for zero disorder. We prove that there exist localized
interfaces with probability one in dimensions , in a
`low-temperature' regime. The proof extends the method of
continuous-to-discrete single- site coarse graining that was previously applied
by the author for a double-well potential to the case of a non-compact image
space. This allows to utilize parts of the renormalization group analysis
developed for the treatment of a contour representation of a related
integer-valued SOS-model in [BoK1]. We show that, for a.e. fixed realization of
the disorder, the infinite volume Gibbs measures then have a representation as
superpositions of massive Gaussian fields with centerings that are distributed
according to the infinite volume Gibbs measures of the disordered
integer-valued SOS-model with exponentially decaying interactions
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