Random Currents and Continuity of Ising Model’s Spontaneous Magnetization
Michael Aizenman, Hugo Duminil-Copin, Vladas Sidoravicius
The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on (Z^d) in (d=3) dimensions. The analysis applies also to higher dimensions, for which the result is already known, and to systems with interactions of power law decay. It allows to conclude similar continuity results for one dimensional systems provided the decay is slower than (1/r^2) (at which the transition is known to be discontinuous). The proof employs in an essential way an extension of Ising model’s random current representation to the model’s infinite volume limit. This extension enables one to reduce the continuity statement to a simple criterion on the decay of correlation in the Gibbs state with free boundary conditions. For reflection positive models, this criterion may be established through the related infrared bound.