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Regularity Properties of Potentials for Joint Measures of Random Spin Systems

By C. Külske


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 for renormalization group transformations.

Year: 2004
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