Geometric analysis of Ising models, Part III

Abstract: The random current representation of the Ising model, along with a related path expansion, has been a source of insight on the stochastic geometric underpinning of the ferromagnetic model's phase structure and critical behavior in different dimensions. This representation is extended in this work to systems with a mild amount of frustration, such as generated by disorder operators and external field of mixed signs. Further examples of the utility of stochastic geometric representations are presented here in the context of the deconfinement transition of the $Z_2$ lattice gauge model -- particularly in three dimensions -- and in streamlined proofs of correlation inequalities with wide-ranging applications.