Christof Kuelske
Analogues of non-Gibbsianness in joint measures of
disordered mean field models
(310K, postscript)
ABSTRACT. It is known that the joint measures on the product of
spin-space and disorder space are very often non-Gibbsian
measures, for lattice systems with quenched disorder,
at low temperature.
Are there reflections of this non-Gibbsianness in the
corresponding mean-field models? We study the continuity
properties of the conditional expectations in finite volume
of the following mean field models:
a) joint measures of random field Ising,
b) joint measures of dilute Ising,
c) decimation of ferromagnetic Ising.
Observing that the conditional expectations are functions of
the empirical mean of the conditionings we look at the large
volume behavior of these functions to discover non-trivial
limiting objects. For a) we find 1) discontinuous dependence
for almost any realization and 2) dependence of the conditional
expectation on the phase. In contrast to that we see continuous
behavior for b) and c), for almost any realization.
This is in complete analogy to the behavior of the corresponding
lattice models in high dimensions. It shows that non-Gibbsian
behavior which seems a genuine lattice phenomenon can be partially
understood already on the level of mean-field models.