Regularity Properties of Potentials for Joint Measures of Random Spin Systems

C. Kuelske

2004, v.10, Issue 1, 75-88


We consider general quenched
disordered lattice spin models on compact local spin spaces with possibly
dependent disorder. We discuss their
corresponding joint measures on the product space of disorder variables
and spin variables in the infinite volume.
These measures often possess pathologies in a low temperature region
reminiscent of renormalization group pathologies in the sense that
they are not Gibbs measures on the product space. Often the joint measures are not
even almost Gibbs, but it is known that there is always a potential
for their conditional expectations that may however only
be summable on a full measure set, and not everywhere.
In this note we complement the picture from the non-pathological side.
We show regularity properties for the potential
in the region of interactions where the joint potential is absolutely summable everywhere.
We prove unicity and Lipschitz-continuity, much in analogy to the two fundamental regularity
theorems proved by van Enter, Fernandez, Sokal for renormalization group transformations.

Keywords: disordered systems, Gibbs measures, non-Gibbsian measures, joint measures, random field model


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