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 here 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 such stochastic geometric representations are presented 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.